y2+y=x3+x2−18725758x+31284762394
|
(homogenize, simplify) |
y2z+yz2=x3+x2z−18725758xz2+31284762394z3
|
(dehomogenize, simplify) |
y2=x3−24268582800x+1459913097258000
|
(homogenize, minimize) |
sage: E = EllipticCurve([0, 1, 1, -18725758, 31284762394])
gp: E = ellinit([0, 1, 1, -18725758, 31284762394])
magma: E := EllipticCurve([0, 1, 1, -18725758, 31284762394]);
oscar: E = elliptic_curve([0, 1, 1, -18725758, 31284762394])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Z⊕Z
magma: MordellWeilGroup(E);
P | h^(P) | Order |
(2207,26680) | 0.39104710520184050807155317343 | ∞ |
(4748,222337) | 0.41672451756626471169244466641 | ∞ |
(−4327,176962), (−4327,−176963), (−3722,222337), (−3722,−222338), (−1726,241798), (−1726,−241799), (−1027,222337), (−1027,−222338), (−397,196612), (−397,−196613), (1514,80041), (1514,−80042), (2207,26680), (2207,−26681), (2447,10999), (2447,−11000), (2603,13612), (2603,−13613), (2648,16537), (2648,−16538), (3278,71662), (3278,−71663), (4748,222337), (4748,−222338), (13673,1526962), (13673,−1526963), (19994,2765878), (19994,−2765879), (31973,5667337), (31973,−5667338), (443648,295486537), (443648,−295486538)
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor: |
N |
= |
444675 | = | 3⋅52⋅72⋅112 |
sage: E.conductor().factor()
|
Discriminant: |
Δ |
= |
−2742040655047461796875 | = | −1⋅37⋅57⋅77⋅117 |
sage: E.discriminant().factor()
|
j-invariant: |
j |
= |
−841995222985990144 | = | −1⋅212⋅3−7⋅5−1⋅7−1⋅11−1⋅3793 |
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 | ≈ | 2.9729903138975793473472962220 |
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | −0.0036313532463127645367316053160 |
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 0.8908239351519434 |
|
Szpiro ratio: |
σm | ≈ | 4.75631432894903 |
|
Analytic rank: |
ran | = | 2
|
|
Mordell-Weil rank: |
r | = | 2
|
gp: [lower,upper] = ellrank(E)
|
Regulator: |
Reg(E/Q) | ≈ | 0.15734750688516797785168778042 |
G = E.gen \\ if available matdet(ellheightmatrix(E,G))
|
Real period: |
Ω | ≈ | 0.14423208227826175698057247894 |
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 | = | 448
= 7⋅22⋅22⋅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 | = | 1 |
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Special value: |
L(2)(E,1)/2! | ≈ | 10.167162234584725927270955376 |
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);
|
10.167162235≈L(2)(E,1)/2!=?#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈121⋅0.144232⋅0.157348⋅448≈10.167162235
# 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
444675.2.a.bd
q−2q2+q3+2q4−2q6+q9+2q12+2q13−4q16−q17−2q18−7q19+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 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.
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
gens = [[1541, 2, 1541, 3], [211, 2, 211, 3], [661, 2, 661, 3], [1387, 2, 1387, 3], [1, 0, 2, 1], [1, 2, 0, 1], [2309, 2, 2308, 3], [1, 1, 2309, 0]]
GL(2,Integers(2310)).subgroup(gens)
Gens := [[1541, 2, 1541, 3], [211, 2, 211, 3], [661, 2, 661, 3], [1387, 2, 1387, 3], [1, 0, 2, 1], [1, 2, 0, 1], [2309, 2, 2308, 3], [1, 1, 2309, 0]];
sub<GL(2,Integers(2310))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
level 2310=2⋅3⋅5⋅7⋅11, index 2, genus 0, and generators
(1541154123),(21121123),(66166123),(1387138723),(1201),(1021),(2309230823),(1230910).
The torsion field K:=Q(E[2310]) is a degree-1839366144000 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/2310Z).
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 |
3 |
split multiplicative |
4 |
148225=52⋅72⋅112 |
5 |
additive |
18 |
17787=3⋅72⋅112 |
7 |
additive |
32 |
3025=52⋅112 |
11 |
additive |
72 |
3675=3⋅52⋅72 |
This curve has no rational isogenies. Its isogeny class 444675bd
consists of this curve only.
The minimal quadratic twist of this elliptic curve is
1155i1, its twist by 385.
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.