| Description | Elliptic curves over Q and other number fields |
| Status | production |
| Contact | John Cremona |
| Code | elliptic_curves |
| Collections | curves, nfcurves, padic_db |
Notes: In good shape over Q (collection curves). Over quadratic and cubic fields (collection nfcurves) complete as far as curves matching Hilbert modular forms go; awaiting addition of curves over more fields of degrees 4-6 to match Hilbert modular forms.
Todo:
- Complete the finding and uploading of curves over totally real fields to match the HMF newforms for fields of degrees >3.
- compute and add p-adic data for curves over Q of conductor > 130000.
- compute more ranks and generators for ecnf
- Content: elliptic curves over Q
- Contributors: John Cremona, Andrew Sutherland, Jeremy Rouse
- Origin: Cremona database, https://github.com/JohnCremona/ecdata
- Extent: complete for conductors up to 380,000 (as of February 2016)
| Field | Description | Type of stored data | Mathematical type | Example of stored data | Remarks |
|---|---|---|---|---|---|
| _id | Mongo id | ObjectId | - | assigned by Mongo; contains creation timestamp | |
| label | Cremona label | string | - | '1225a2' | |
| lmfdb_label | LMFDB label | string | - | '1225.a2' | |
| conductor | Conductor | int | N | 1225 | |
| iso | Cremona isogeny class code | string | - | '11a' | |
| lmfdb_iso | LMFDB isogeny class code | string | - | '11.a' | |
| iso_nlabel | numerical version of the LMFDB isogeny class label | int | Z | 0 | |
| number | Cremona curve number within its class | int | N | 2 | |
| lmfdb_number | LMFDB curve number within its class | int | N | 2 | |
| ainvs | a-invariants (coefficients of minimal reduced Weierstass model) | list of 5 strings representing integers | Z^5 | ['0', '1', '1', '10617', '75394'] | |
| xainvs | a-invariants (coefficients of minimal reduced Weierstass model) | string representing list of 5 integers | Z^5 | '[0, 1, 1, 10617, 75394]' | |
| jinv | j-invariant | string representing a rational | Q | '-4096/11' | |
| signD | sign of Discriminant | int | Z | -1 | in {-1,+1} |
| cm | CM code | int | Z | 0 (for no CM), or a negative discriminant | in {0, -3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163} |
| rank | rank | int | N_0 | 0 | May be missing |
| torsion | torsion order | int | Z | 1 | |
| torsion_structure | invariants of torsion subgroup | list of at most 2 strings representing ints | N^t (0≤t&le2) | ['3'] | |
| torsion_generators | generators of torsion subgroup | list of strings representing points | A^2(Q)^t (0≤t&le2) | ['(5, 5)'] | |
| x-coordinates_of_integral_points | x-coordinates of integral points | string representing list of integers | Z^k | '[5,16]' | |
| gens | generators of infinite order | list of strings representing points | P^2(Q)^k (k≥0) | ['(0:0:1)'] | May be missing |
| heights | heights of generators | list of floats | R^k (k≥0) | [0.4754476141654406, 2.4031247402073275] | May be missing |
| regulator | regulator | float | R | 1.0 | May be missing; approximate if rank>0 |
| tamagawa_product | Tamagawa product | int | N | 4 | |
| special_value | special value of r'th derivative of L-function (divided by r!) | float | R | 1.490882041449698 | approximate |
| real_period | real period | float | R | 0.3727205103624245 | approximate |
| degree | degree of modular parametrization | int | N | 1984 | |
| non-surjective_primes | primes p for which the mod p Galois representation is not surjective | list of ints | N^k (k≥0) | [5] | |
| non-maximal_primes | primes p for which the mod p Galois representation is not maximal | list of ints | N^k (k≥0) | [5] | |
| galois_images | Sutherland codes for the images of the mod p Galois representations for the non-surjective primes | list of strings | - | ['5B'] | Sutherland notation; for CM curves, only primes<100 |
| mod-p_images | Sutherland codes for the images of the mod p Galois representations for the non-maximal primes | list of strings | - | ['5B'] | Sutherland notation |
| 2adic_label | Rouse label of the associated modular curve (None for CM curves) | string | - | 'X225g' | based on Rouse, Zureik-Brown classification |
| 2adic_index | index in GL(2,Z2) of the 2-adic representation (or 0 for CM curves) | int | N | 1 | |
| 2adic_log_level | the smallest n such that the image contains the kernel of reduction modulo 2^n (or None for CM curves) | int | N_0 | 1 | |
| 2adic_gens | list of matrices in GL(2,Z/2^nZ) generating the image (None for CM curves) | list of lists of 4 ints | GL(2,Z)^k (k≥0) | [[5,0,0,5],[5,5,0,1],[5,5,0,3]] | |
| isogeny_matrix | matrix of isogeny degrees for class | list of lists of ints | M_k(N) (k≥0) | [[1,5,25],[5,1,5],[25,5,1]] | |
| isogeny_degrees | degrees of cyclic isogenies | list of ints | N^k (k≥0) | [1,5,25] | |
| class_deg | maximal degree of a cyclic isogeny in the class | int | N | 25 | |
| class_size | size of the isogeny class | int | N | 3 | |
| sha_an | analytic order of Sha | float | R | 9.0 | approximate unless rank<2 |
| sha | analytic order of sha | int | N | 9 | rounded value of sha_an |
| sha_primes | primes dividing sha | list of ints | N^k (k≥0) | [2] | |
| torsion_primes | primes dividing torsion | list of ints | N^k (k≥0) | [2,3] | |
| local_data | reduction data at bad primes | list of dicts, one per prime, each with keys 'p' (value:int), 'ord_cond' (value: int), 'ord_disc' (value: int), 'ord_den_j' (value: int), 'red' (value: int), 'cp' (value: int), 'kod' (value: string) | [{'cp': 1, 'kod': '\\( I_{1} \\)', 'ord_cond': 1, 'ord_den_j': 1, 'ord_disc': 1, 'p': 11, 'red': 1}] | ||
| min_quad_twist | minimal quadratic twist | dict with keys 'label' (value:string) and 'disc' (value: int) | N | {'disc': 1, 'label': '11a2'} | |
| aplist | Traces of Frobenius | list of 25 ints | Z^25 | [0, 1, -1, ..., 2] | a_p for p<100 |
| anlist | L-series coefficients | list of 20 ints | Z^20 | [0, 1, -1, ..., 2] | a_n for 0<=np<20 |
| iwdata | Iwasawa invariants | dictionary with keys ints, values lists of ints | Z^20 | {u'11': [1, 0], u'3': [0, 0], u'2': [0, 1, 0], u'5': [0, 1]} | keys are primes, including all bad multiplicative primes and all primes up to some bound |
| iwp0 | Iwasawa prime | int | N | 7 | if nonzero, a prime p0 such that lambda=mu=0 for all good p>=p0 |
Index information on collection curves:
- {'_id': 1} (created by mongo)
- {'rank': 1, 'number': 1} (for searching and stats)
- {'number': 1} (for searching and stats)
- {'conductor': 1, 'iso_nlabel': 1, 'lmfdb_number': 1} (for sorting)
- {'non-surjective_primes': 1} (for searching)
- {'non-maximal_primes': 1} (for searching)
- {'cm': 1} (for searching)
- {'conductor': 1} (for searching)
- {'lmfdb_label': 1,'number': 1} (for searching)
- {'lmfdb_iso': 1} (for searching)
- {'sha': 1} (for searching)
- {'lmfdb_label': 1} (for searching)
- {'rank': 1} (for searching)
- {'label': 1} (for searching)
- {'jinv': 1} (for searching)
- {'torsion_structure': 1} (for searching)
- {'iso': 1} (for searching)
- {'torsion': 1} (for searching)
- {'label': 1, 'number': 1} (for searching)
- {'xainvs': 1} (for searching)
- curves.rand (auxilliary collection used for random objection access)
- curves.stats (auxilliary collection of statistics)
- Content: p-adic regulators for elliptic curves over Q
- Contributors: unkown
- Origin: unknown
- Extent: primes p with 3,p<100 of good ordinary reduction, for curves of conductor up to 130,000 only (last updated in 2010)
| Field | Description | Type of stored data | Mathematical type | Example of stored data | Remarks |
|---|---|---|---|---|---|
| _id | Mongo id | ObjectId | assigned by Mongo; contains creation timestamp | ||
| lmfdb_iso | LMFDB label of isogeny class | string | - | '58.a' | |
| p | prime | int | N (prime) | 97 | |
| prec | p-adic precision | int | N | 20 | |
| unit | unit factor of regulator | string representing integer | Z_p (mod p^N) | '8471152617139064438417376357679138234' | |
| val | valuation of p-adic regulator | int | N_0 | 1 |
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Content: elliptic curves over number fields other than Q
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Contributors: John Cremona, Alyson Deines, Steve Donelly, Paul Gunnells, Warren Moore, Haluk Sengun, John Voight, Dan Yasaki.
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Extent: contains curves over several totally real fields (of degrees up to 6) and a few imaginary quadratic fields, in each case complete up to some conductor norm bound
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Updated collection and description 22 July 2016.
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Explanation of data fields representing elements of the field, including points:
- Each field of degree d has a distinguished generator w.
- A rational number is represented as a string.
- A field element (NFelt) is usually represented as an NFelt-string, being d rational-strings, representing the coordinates with respect to the w-power basis, joined by ","; or in some contexts as a string representing the element as a polynomial in the field generator w.
- A point (in projective coordinates) is represented as a point-string: 3 NFelt-strings surrounded by "[","]", joined with ",", the whole surrounded again by "[","]".
- An ideal is represented as an ideal-string [N,a,alpha] where N is the norm, a the smallest positive integer and alpha a second generator expressed as a polynomial in w.
| Field | Description | Type of stored data | Mathematical type | Example of stored data | Remarks | |
|---|---|---|---|---|---|---|
| _id | Mongo id | ObjectId | - | assigned by Mongo; contains creation timestamp | ||
| field_label | Base field label | string | '2.0.8.1' | |||
| degree | Base field degree | int | N | 2 | ||
| signature | Base field signature | list of 2 ints | N_0^2 | [0, 1] | ||
| abs_disc | absolute value of discriminant of base field | int | N | 8 | ||
| label | full label | string | - | '2.0.8.1-[3618,1146,3]-e2' | ||
| short_label | short label (excludes field) | string | - | '[3618,1146,3]-e2' | ||
| class_label | full label of isogeny class | string | - | '2.0.8.1-[3618,1146,3]-e' | ||
| short_class_label | short label of isogeny class (excludes field) | string | - | '2.0.8.1-[3618,1146,3]-e' | ||
| conductor_label | condcutor label | string | - | '[3618,1146,3]' or '37.1' | ||
| iso_label | isogeny class label | string | - | 'e' | base 26 representation of isogeny class index | |
| iso_nlabel | isogeny class index | int | N_0 | 4 | ||
| conductor_ideal | data defining the conductor | ideal-string | - | '[13931,13931,-25*w^2+11*w]' | representation generators | |
| conductor_norm | conductor norm | int | N | 3618 | ||
| number | index of curve in isogeny class | int | N | 2 | starts at 1 | |
| isogeny_matrix | Isogeny matrix | list of list of ints (degrees) | M_k(N) (k≥0) | [[1, 2], [2, 1]] | ||
| isogeny_degrees | degrees of cyclic isogenies | list of ints | N^k (k≥0) | [1,5,25] | ||
| class_deg | maximal degree of a cyclic isogeny in the class | int | N | 25 | ||
| class_size | size of the isogeny class | int | N | 3 | ||
| ainvs | a-invariants | string | '0,0,1;0,0,0;1,1,1;0,2,-2;1,-1,-1' | 5 Weierstrass coefficients as NFelt-strings, joined by ";" | ||
| jinv | j-invariant | string | ['288857903821/4771277298', '7556047939133/9542554596'] | NFelt-string | ||
| analytic_rank | analytic rank | int | N_0 | 0 | ||
| rank | rank | int | N_0 | 0 | ||
| rank_bounds | lower and upper rank bounds | list of 2 ints | N_0^2 | [0, 0] | ||
| sha_an | analytic order of Sha | int | N | 1 | rounded float | |
| ngens | number of stored generators | int | N_0 | 0 | ||
| gens | generators of infinite order | list of point-strings | P^2(K)^r (0≤r≤rank) | ['[[0,-1,1],[0,0,0],[1,0,0]]', '[[0,-1,0],[-2,-1,1],[1,0,0]]'] | ||
| torsion_order | torsion order | int | N | 6 | ||
| torsion_structure | invariants of torsion subgroup | list of at most 2 ints | N^t (0≤t&le2) | [3, 3] | ||
| torsion_gens | torsion generators | list of point-strings | P^2(K)^t (0≤t&le2) | ['[[0,0,0],[0,1,-1],[1,0,0]]', '[[1/2,-3/4,0],[-5/8,1/8,1/2],[1,0,0]]'] | ||
| q_curve | Q-curve flag | boolean | {True, False} | False | ||
| base_change | labels of base change source curves | list of strings | - | ['4032.k2', '63.a2'] | ||
| cm | CM code | int | Z | 0 (for no CM), or a negative discriminant | ||
| local_data | List of local data at bad primes | {'p','normp','red','kod',ord_cond','ord_disc','ord_den_j'}^k | [{'cp': 1, 'kod': '\\( I_{27} \\)', 'normp': '2', 'ord_cond': 1, 'ord_den_j': 27, 'ord_disc': 27, 'p': '[2,2,-w-10]', 'red': -1}] | 'p': ideal_string (prime ideal), 'normp':int (its norm), 'kod': string (Kodaira symbol), 'red': {-1,0,1} (reduction type), ord_cond:int (conductor exponent), ord_disc: int (discriminant exponent), ord_den_:int (j-invariants denominator exponent) | ||
| minD | minimal discriminant ideal | ideal-string | Ideal | '[1399489,199927,101*w^2-44*w-121] | ||
| heights | heights of generators | list of floats | R^r | [0.2815779492939666, 0.25465400108973096] | ||
| reg | regulator | float | R | 0.01024588468931388 | ||
| non_min_p | Non-minimal primes | list of ideal-strings | {ideals}^k | [] | ||
| equation | Weierstrass equation | string | '\\( y^2 + x y + y = x^{3} + a x^{2} + \\left(-30047 a - 303287\\right) x - 9341927 a - 94305014 \\) | |||
| non-surjective_primes | primes p for which the mod p Galois representation is not surjective | list of ints | N^k (k≥0) | [5] | ||
| galois_images | Sutherland codes for the images of the mod p Galois representations for the non-surjective primes | list of strings | - | ['5B'] | Sutherland notation; for CM curves, only primes<100 |
Index information for collection nfcurves
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{'_id': 1} (created by mongo)
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{'field_label': 1}
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{'degree': 1}
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{'number': 1}
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{'label': 1}
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{'field_label':1, 'conductor_norm':1, 'conductor_label':1, 'iso_nlabel':1,'number':1}
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nfcurves.rand (auxilliary collection used for random objection access)
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nfcurves.stats (auxilliary collection of statistics)