sage:E = EllipticCurve([0, 0, 0, -14475, -670250])
gp:E = ellinit([0, 0, 0, -14475, -670250])
magma:E := EllipticCurve([0, 0, 0, -14475, -670250]);
oscar:E = elliptic_curve([0, 0, 0, -14475, -670250])
sage:E.short_weierstrass_model()
magma:WeierstrassModel(E);
oscar:short_weierstrass_model(E)
Z⊕Z/2Z
magma:MordellWeilGroup(E);
P | h^(P) | Order |
(−69,4) | 1.6659303477317317899848332873 | ∞ |
(−70,0) | 0 | 2 |
(−70,0), (−69,±4), (155,±900), (714,±18788)
sage:E.integral_points()
magma:IntegralPoints(E);
Invariants
Conductor: |
N |
= |
1800 | = | 23⋅32⋅52 |
sage:E.conductor().factor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
oscar:conductor(E)
|
Discriminant: |
Δ |
= |
34992000000 | = | 210⋅37⋅56 |
sage:E.discriminant().factor()
gp:E.disc
magma:Discriminant(E);
oscar:discriminant(E)
|
j-invariant: |
j |
= |
328756228 | = | 22⋅3−1⋅1933 |
sage:E.j_invariant().factor()
gp:E.j
magma:jInvariant(E);
oscar:j_invariant(E)
|
Endomorphism ring: |
End(E) | = | Z |
Geometric endomorphism ring: |
End(EQ) |
= |
Z
(no potential complex multiplication)
|
sage:E.has_cm()
magma:HasComplexMultiplication(E);
|
Sato-Tate group: |
ST(E) | = | SU(2) |
Faltings height: |
hFaltings | ≈ | 1.0552464047512204150476076690 |
gp:ellheight(E)
magma:FaltingsHeight(E);
oscar:faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | −0.87640134626650570913142138396 |
magma:StableFaltingsHeight(E);
oscar:stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 1.0561716345991692 |
|
Szpiro ratio: |
σm | ≈ | 5.38375186773158 |
|
Analytic rank: |
ran | = | 1
|
sage:E.analytic_rank()
gp:ellanalyticrank(E)
magma:AnalyticRank(E);
|
Mordell-Weil rank: |
r | = | 1
|
sage:E.rank()
gp:[lower,upper] = ellrank(E)
magma:Rank(E);
|
Regulator: |
Reg(E/Q) | ≈ | 1.6659303477317317899848332873 |
sage:E.regulator()
gp:G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma:Regulator(E);
|
Real period: |
Ω | ≈ | 0.43525887000349523419004959539 |
sage:E.period_lattice().omega()
gp:if(E.disc>0,2,1)*E.omega[1]
magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
Tamagawa product: |
∏pcp | = | 16
= 2⋅22⋅2
|
sage:E.tamagawa_numbers()
gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma:TamagawaNumbers(E);
oscar:tamagawa_numbers(E)
|
Torsion order: |
#E(Q)tor | = | 2 |
sage:E.torsion_order()
gp:elltors(E)[1]
magma:Order(TorsionSubgroup(E));
oscar:prod(torsion_structure(E)[1])
|
Special value: |
L′(E,1) | ≈ | 2.9004438426329738350460360566 |
sage:r = E.rank();
E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp:[r,L1r] = ellanalyticrank(E); L1r/r!
magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
Analytic order of Ш: |
Шan |
≈ |
1
(rounded)
|
sage:E.sha().an_numerical()
magma:MordellWeilShaInformation(E);
|
2.900443843≈L′(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈221⋅0.435259⋅1.665930⋅16≈2.900443843
sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([0, 0, 0, -14475, -670250]); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
E := EllipticCurve([0, 0, 0, -14475, -670250]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
Modular form
1800.2.a.m
q−4q11+2q13+2q17−4q19+O(q20)
sage:E.q_eigenform(20)
gp:\\ actual modular form, use for small N
[mf,F] = mffromell(E)
Ser(mfcoefs(mf,20),q)
\\ or just the series
Ser(ellan(E,20),q)*q
magma:ModularForm(E);
For more coefficients, see the Downloads section to the right.
This elliptic curve is not semistable.
There
are 3 primes p
of bad reduction:
sage:E.local_data()
gp:ellglobalred(E)[5]
magma:[LocalInformation(E,p) : p in BadPrimes(E)];
oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
The ℓ-adic Galois representation has maximal image
for all primes ℓ except those listed in the table below.
sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage:gens = [[5, 4, 236, 237], [143, 0, 0, 239], [225, 16, 224, 17], [1, 0, 16, 1], [15, 2, 142, 227], [61, 160, 60, 121], [1, 16, 0, 1], [106, 115, 95, 126], [64, 235, 45, 14]]
GL(2,Integers(240)).subgroup(gens)
magma:Gens := [[5, 4, 236, 237], [143, 0, 0, 239], [225, 16, 224, 17], [1, 0, 16, 1], [15, 2, 142, 227], [61, 160, 60, 121], [1, 16, 0, 1], [106, 115, 95, 126], [64, 235, 45, 14]];
sub<GL(2,Integers(240))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
level 240=24⋅3⋅5, index 192, genus 1, and generators
(52364237),(14300239),(2252241617),(11601),(151422227),(6160160121),(10161),(10695115126),(644523514).
The torsion field K:=Q(E[240]) is a degree-2949120 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/240Z).
The table below list all primes ℓ for which the Serre invariants associated to the mod-ℓ Galois representation are exceptional.
ℓ |
Reduction type |
Serre weight |
Serre conductor |
2 |
additive |
2 |
225=32⋅52 |
3 |
additive |
8 |
200=23⋅52 |
5 |
additive |
14 |
72=23⋅32 |
gp:ellisomat(E)
This curve has non-trivial cyclic isogenies of degree d for d=
2, 4 and 8.
Its isogeny class 1800s
consists of 6 curves linked by isogenies of
degrees dividing 8.
The minimal quadratic twist of this elliptic curve is
24a3, its twist by −15.
The number fields K of degree less than 24 such that
E(K)tors is strictly larger than E(Q)tors
≅Z/2Z
are as follows:
We only show fields where the torsion growth is primitive.
For fields not in the database, click on the degree shown to reveal the defining polynomial.
An entry - indicates that the invariants are not computed because the reduction is additive.
p-adic regulators
p-adic regulators are not yet computed for curves that are not Γ0-optimal.