y2+xy=x3−1480x+94400
|
(homogenize, simplify) |
y2z+xyz=x3−1480xz2+94400z3
|
(dehomogenize, simplify) |
y2=x3−1918107x+4410080694
|
(homogenize, minimize) |
sage: E = EllipticCurve([1, 0, 0, -1480, 94400])
gp: E = ellinit([1, 0, 0, -1480, 94400])
magma: E := EllipticCurve([1, 0, 0, -1480, 94400]);
oscar: E = elliptic_curve([1, 0, 0, -1480, 94400])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Z/7Z
magma: MordellWeilGroup(E);
(−40,320), (−40,−280), (20,260), (20,−280), (110,1070), (110,−1180)
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor: |
N |
= |
5010 | = | 2⋅3⋅5⋅167 |
sage: E.conductor().factor()
|
Discriminant: |
Δ |
= |
−3652290000000 | = | −1⋅27⋅37⋅57⋅167 |
sage: E.discriminant().factor()
|
j-invariant: |
j |
= |
−3652290000000358531401121921 | = | −1⋅2−7⋅3−7⋅5−7⋅193⋅167−1⋅37393 |
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 | ≈ | 1.0927379440021322213072649238 |
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | 1.0927379440021322213072649238 |
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 0.9611687724003037 |
|
Szpiro ratio: |
σm | ≈ | 4.276972863408639 |
|
Analytic rank: |
ran | = | 0
|
|
Mordell-Weil rank: |
r | = | 0
|
gp: [lower,upper] = ellrank(E)
|
Regulator: |
Reg(E/Q) | = | 1 |
G = E.gen \\ if available matdet(ellheightmatrix(E,G))
|
Real period: |
Ω | ≈ | 0.67206428555649961536038156913 |
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 | = | 343
= 7⋅7⋅7⋅1
|
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 | = | 7 |
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Special value: |
L(E,1) | ≈ | 4.7044499988954973075226709839 |
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
(exact)
|
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
4.704449999≈L(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈721⋅0.672064⋅1.000000⋅343≈4.704449999
# 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
5010.2.a.h
q+q2+q3+q4+q5+q6+q7+q8+q9+q10−2q11+q12+q14+q15+q16+4q17+q18−q19+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 semistable.
There
are 4 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 = [[8, 7, 35063, 140274], [117608, 7, 107513, 140274], [8, 7, 70133, 140274], [1, 14, 0, 1], [46768, 7, 46753, 140274], [105211, 70154, 0, 65131], [84176, 7, 84161, 140274], [1, 0, 14, 1], [8, 5, 91, 57], [140267, 14, 140266, 15]]
GL(2,Integers(140280)).subgroup(gens)
Gens := [[8, 7, 35063, 140274], [117608, 7, 107513, 140274], [8, 7, 70133, 140274], [1, 14, 0, 1], [46768, 7, 46753, 140274], [105211, 70154, 0, 65131], [84176, 7, 84161, 140274], [1, 0, 14, 1], [8, 5, 91, 57], [140267, 14, 140266, 15]];
sub<GL(2,Integers(140280))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
level 140280=23⋅3⋅5⋅7⋅167, index 96, genus 2, and generators
(8350637140274),(1176081075137140274),(8701337140274),(10141),(46768467537140274),(10521107015465131),(84176841617140274),(11401),(891557),(1402671402661415).
The torsion field K:=Q(E[140280]) is a degree-574559373376880640 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/140280Z).
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 |
split multiplicative |
4 |
2505=3⋅5⋅167 |
3 |
split multiplicative |
4 |
1670=2⋅5⋅167 |
5 |
split multiplicative |
6 |
1002=2⋅3⋅167 |
7 |
good |
2 |
167 |
167 |
nonsplit multiplicative |
168 |
30=2⋅3⋅5 |
This curve has non-trivial cyclic isogenies of degree d for d=
7.
Its isogeny class 5010h
consists of 2 curves linked by isogenies of
degree 7.
This elliptic curve is its own minimal quadratic twist.
The number fields K of degree less than 24 such that
E(K)tors is strictly larger than E(Q)tors
≅Z/7Z
are as follows:
[K:Q] |
K |
E(K)tors |
Base change curve |
3 |
3.1.20040.1 |
Z/14Z |
not in database
|
6 |
6.0.8048096064000.1 |
Z/2Z⊕Z/14Z |
not in database
|
8 |
deg 8 |
Z/21Z |
not in database
|
12 |
deg 12 |
Z/28Z |
not in database
|
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.
All Iwasawa λ and μ-invariants for primes p≥11 of good reduction are zero.
p-adic regulators
All p-adic regulators are identically 1 since the rank is 0.