Properties

Label 37845d1
Conductor 3784537845
Discriminant 6.504×1012-6.504\times 10^{12}
j-invariant 115 -\frac{1}{15}
CM no
Rank 00
Torsion structure Z/2Z\Z/{2}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x3x2158x122668y^2+xy+y=x^3-x^2-158x-122668 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x3x2z158xz2122668z3y^2z+xyz+yz^2=x^3-x^2z-158xz^2-122668z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x32523x7853258y^2=x^3-2523x-7853258 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 1, -158, -122668])
 
gp: E = ellinit([1, -1, 1, -158, -122668])
 
magma: E := EllipticCurve([1, -1, 1, -158, -122668]);
 
oscar: E = elliptic_curve([1, -1, 1, -158, -122668])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z/2Z\Z/{2}\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(51,26)(51, -26)0022

Integral points

(51,26) \left(51, -26\right) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  37845 37845  = 3252923^{2} \cdot 5 \cdot 29^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  6504393015135-6504393015135 = 1375296-1 \cdot 3^{7} \cdot 5 \cdot 29^{6}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  115 -\frac{1}{15}  = 13151-1 \cdot 3^{-1} \cdot 5^{-1}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 1.13752930668450276242843001601.1375293066845027624284300160
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.0954247526427890968608286186-1.0954247526427890968608286186
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.19807684405159481.1980768440515948
Szpiro ratio: σm\sigma_{m} ≈ 3.50605885380313563.5060588538031356

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.342270258996964157328021118330.34227025899696415732802111833
comment: Real Period
 
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\prod_{p}c_p = 8 8  = 2212 2^{2}\cdot1\cdot2
comment: Tamagawa numbers
 
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\#E(\Q)_{\mathrm{tor}} = 22
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 0.684540517993928314656042236650.68454051799392831465604223665
comment: Special L-value
 
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{}_{\mathrm{an}}  =  11    (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

0.684540518L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.3422701.0000008220.684540518\displaystyle 0.684540518 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.342270 \cdot 1.000000 \cdot 8}{2^2} \approx 0.684540518

# 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 invariants

Modular form   37845.2.a.d

qq2q4q5+3q8+q104q112q13q16+2q174q19+O(q20) q - q^{2} - q^{4} - q^{5} + 3 q^{8} + q^{10} - 4 q^{11} - 2 q^{13} - q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20}) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ 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.

Modular degree: 50176
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
33 44 I1I_{1}^{*} additive -1 2 7 1
55 11 I1I_{1} nonsplit multiplicative 1 1 1 1
2929 22 I0I_0^{*} additive 1 2 6 0

comment: Local data
 
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)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
22 2B 32.48.0.1

comment: mod p Galois image
 
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

gens = [[1, 0, 32, 1], [1, 32, 0, 1], [5, 28, 68, 381], [6903, 8642, 11194, 5743], [1654, 12963, 4205, 12268], [4319, 0, 0, 13919], [23, 18, 11358, 11915], [608, 13891, 10933, 318], [13889, 32, 13888, 33], [1, 12992, 13050, 4351]]
 
GL(2,Integers(13920)).subgroup(gens)
 
Gens := [[1, 0, 32, 1], [1, 32, 0, 1], [5, 28, 68, 381], [6903, 8642, 11194, 5743], [1654, 12963, 4205, 12268], [4319, 0, 0, 13919], [23, 18, 11358, 11915], [608, 13891, 10933, 318], [13889, 32, 13888, 33], [1, 12992, 13050, 4351]];
 
sub<GL(2,Integers(13920))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 13920=253529 13920 = 2^{5} \cdot 3 \cdot 5 \cdot 29 , index 768768, genus 1313, and generators

(10321),(13201),(52868381),(69038642111945743),(165412963420512268),(43190013919),(23181135811915),(6081389110933318),(13889321388833),(112992130504351)\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 6903 & 8642 \\ 11194 & 5743 \end{array}\right),\left(\begin{array}{rr} 1654 & 12963 \\ 4205 & 12268 \end{array}\right),\left(\begin{array}{rr} 4319 & 0 \\ 0 & 13919 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 11358 & 11915 \end{array}\right),\left(\begin{array}{rr} 608 & 13891 \\ 10933 & 318 \end{array}\right),\left(\begin{array}{rr} 13889 & 32 \\ 13888 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 12992 \\ 13050 & 4351 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[13920])K:=\Q(E[13920]) is a degree-80461430784008046143078400 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/13920Z)\GL_2(\Z/13920\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
33 additive 88 4205=5292 4205 = 5 \cdot 29^{2}
55 nonsplit multiplicative 66 7569=32292 7569 = 3^{2} \cdot 29^{2}
2929 additive 422422 45=325 45 = 3^{2} \cdot 5

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 4, 8 and 16.
Its isogeny class 37845d consists of 8 curves linked by isogenies of degrees dividing 16.

Twists

The minimal quadratic twist of this elliptic curve is 15a8, its twist by 87-87.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} Z/2Z\cong \Z/{2}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(15)\Q(\sqrt{-15}) Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
22 Q(145)\Q(\sqrt{145}) Z/4Z\Z/4\Z not in database
22 Q(87)\Q(\sqrt{-87}) Z/4Z\Z/4\Z not in database
44 Q(15,87)\Q(\sqrt{-15}, \sqrt{-87}) Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
44 Q(3,29)\Q(\sqrt{-3}, \sqrt{29}) Z/8Z\Z/8\Z not in database
44 Q(5,87)\Q(\sqrt{5}, \sqrt{-87}) Z/8Z\Z/8\Z not in database
88 8.0.2062431396000000.8 Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
88 8.4.2062431396000000.5 Z/8Z\Z/8\Z not in database
88 8.0.35806100625.4 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 8.0.895152515625.3 Z/16Z\Z/16\Z not in database
88 8.0.895152515625.1 Z/16Z\Z/16\Z not in database
88 deg 8 Z/6Z\Z/6\Z not in database
1616 deg 16 Z/4ZZ/4Z\Z/4\Z \oplus \Z/4\Z not in database
1616 deg 16 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
1616 deg 16 Z/16Z\Z/16\Z not in database
1616 16.0.801298026229765869140625.1 Z/2ZZ/16Z\Z/2\Z \oplus \Z/16\Z not in database
1616 deg 16 Z/32Z\Z/32\Z not in database
1616 deg 16 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1616 deg 16 Z/12Z\Z/12\Z not in database
1616 deg 16 Z/12Z\Z/12\Z 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.

Iwasawa invariants

pp 2 3 5 29
Reduction type ord add nonsplit add
λ\lambda-invariant(s) 3 - 0 -
μ\mu-invariant(s) 0 - 0 -

All Iwasawa λ\lambda and μ\mu-invariants for primes p3p\ge 3 of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.