y2+xy=x3+x2+1244967x+403409637
|
(homogenize, simplify) |
y2z+xyz=x3+x2z+1244967xz2+403409637z3
|
(dehomogenize, simplify) |
y2=x3+1613476557x+18797277872142
|
(homogenize, minimize) |
sage: E = EllipticCurve([1, 1, 0, 1244967, 403409637])
gp: E = ellinit([1, 1, 0, 1244967, 403409637])
magma: E := EllipticCurve([1, 1, 0, 1244967, 403409637]);
oscar: E = elliptic_curve([1, 1, 0, 1244967, 403409637])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Z⊕Z/2Z
magma: MordellWeilGroup(E);
P | h^(P) | Order |
(237,26562) | 1.8130333167870135835455872545 | ∞ |
(−302,151) | 0 | 2 |
(−302,151), (227,26302), (227,−26529), (237,26562), (237,−26799), (2514,138135), (2514,−140649)
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor: |
N |
= |
177870 | = | 2⋅3⋅5⋅72⋅112 |
sage: E.conductor().factor()
|
Discriminant: |
Δ |
= |
−193618722013942579200 | = | −1⋅216⋅34⋅52⋅77⋅116 |
sage: E.discriminant().factor()
|
j-invariant: |
j |
= |
9289728001023887723039 | = | 2−16⋅3−4⋅5−2⋅7−1⋅100793 |
sage: E.j_invariant().factor()
|
Endomorphism ring: |
End(E) | = | Z |
Geometric endomorphism ring: |
End(EQ) |
= |
Z
(no potential complex multiplication)
|
magma: HasComplexMultiplication(E);
|
Sato-Tate group: |
ST(E) | = | SU(2) |
Faltings height: |
hFaltings | ≈ | 2.5800377667025143527022221652 |
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | 0.40813505577567242811857400450 |
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 0.999810038071446 |
|
Szpiro ratio: |
σm | ≈ | 4.443569582769581 |
|
Analytic rank: |
ran | = | 1
|
|
Mordell-Weil rank: |
r | = | 1
|
gp: [lower,upper] = ellrank(E)
|
Regulator: |
Reg(E/Q) | ≈ | 1.8130333167870135835455872545 |
G = E.gen \\ if available matdet(ellheightmatrix(E,G))
|
Real period: |
Ω | ≈ | 0.11691591723079752629073206806 |
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 | = | 64
= 2⋅2⋅2⋅2⋅22
|
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 |
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Special value: |
L′(E,1) | ≈ | 3.3915592512343806625409979972 |
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);
|
3.391559251≈L′(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈221⋅0.116916⋅1.813033⋅64≈3.391559251
# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve(%s); 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)))
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
E := EllipticCurve(%s); 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
177870.2.a.k
q−q2−q3+q4−q5+q6−q8+q9+q10−q12−2q13+q15+q16+2q17−q18+4q19+O(q20)
\\ 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
For more coefficients, see the Downloads section to the right.
This elliptic curve is not semistable.
There
are 5 primes p
of bad reduction:
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)];
gens = [[8823, 13442, 32362, 20175], [1, 0, 32, 1], [29591, 26906, 5478, 34475], [36929, 32, 36928, 33], [12343, 26906, 17622, 3851], [5, 28, 68, 381], [29248, 26851, 2453, 10878], [5281, 30272, 11594, 34079], [1, 32, 0, 1], [6719, 0, 0, 36959], [23, 18, 34398, 34955]]
GL(2,Integers(36960)).subgroup(gens)
Gens := [[8823, 13442, 32362, 20175], [1, 0, 32, 1], [29591, 26906, 5478, 34475], [36929, 32, 36928, 33], [12343, 26906, 17622, 3851], [5, 28, 68, 381], [29248, 26851, 2453, 10878], [5281, 30272, 11594, 34079], [1, 32, 0, 1], [6719, 0, 0, 36959], [23, 18, 34398, 34955]];
sub<GL(2,Integers(36960))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
level 36960=25⋅3⋅5⋅7⋅11, index 768, genus 13, and generators
(8823323621344220175),(13201),(2959154782690634475),(36929369283233),(1234317622269063851),(56828381),(2924824532685110878),(5281115943027234079),(10321),(67190036959),(23343981834955).
The torsion field K:=Q(E[36960]) is a degree-313918488576000 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/36960Z).
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 |
nonsplit multiplicative |
4 |
5929=72⋅112 |
3 |
nonsplit multiplicative |
4 |
59290=2⋅5⋅72⋅112 |
5 |
nonsplit multiplicative |
6 |
35574=2⋅3⋅72⋅112 |
7 |
additive |
32 |
3630=2⋅3⋅5⋅112 |
11 |
additive |
62 |
1470=2⋅3⋅5⋅72 |
This curve has non-trivial cyclic isogenies of degree d for d=
2, 4, 8 and 16.
Its isogeny class 177870il
consists of 8 curves linked by isogenies of
degrees dividing 16.
The minimal quadratic twist of this elliptic curve is
210e1, its twist by 77.
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.
No Iwasawa invariant data is available for this curve.
p-adic regulators
p-adic regulators are not yet computed for curves that are not Γ0-optimal.