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)

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

Mordell-Weil group structure

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

Copy content comment:Mordell-Weil group
 
Copy content 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

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

Invariants

Conductor: NN  =  37845 37845  = 3252923^{2} \cdot 5 \cdot 29^{2}
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Discriminant: Δ\Delta  =  6504393015135-6504393015135 = 1375296-1 \cdot 3^{7} \cdot 5 \cdot 29^{6}
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: jj  =  115 -\frac{1}{15}  = 13151-1 \cdot 3^{-1} \cdot 5^{-1}
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content 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)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 1.13752930668450276242843001601.1375293066845027624284300160
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.0954247526427890968608286186-1.0954247526427890968608286186
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content 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
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: Ω\Omega ≈ 0.342270258996964157328021118330.34227025899696415732802111833
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content 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
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 22
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 0.684540517993928314656042236650.68454051799392831465604223665
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  =  11    (exact)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

0.684540518L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.3422701.0000008220.684540518\begin{aligned} 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\end{aligned}

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, -1, 1, -158, -122668]); 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)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, -1, 1, -158, -122668]); 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

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content 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
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

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

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content 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

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

Copy content comment:Adelic image of Galois representation
 
Copy content sage: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)
 
Copy content magma: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

Copy content comment:Isogenies
 
Copy content 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.