Properties

Label 76664.e2
Conductor $76664$
Discriminant $-1.180\times 10^{22}$
j-invariant \( -\frac{2065624967846736}{17960084863} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -23063543, -42951211350])
 
gp: E = ellinit([0, 0, 0, -23063543, -42951211350])
 
magma: E := EllipticCurve([0, 0, 0, -23063543, -42951211350]);
 

\(y^2=x^3-23063543x-42951211350\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(\frac{127156062763397936272870903102671675870733181362431}{22578328320588889017864467506265898599882904025}, \frac{257985237337747985166856070561480160031698524821868962085382352256726221296}{3392639201559173383295635160533564428461523936307825032938591546676125}\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $115.07723816905286890066883028$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(5550, 0\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(5550, 0\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 76664 \)  =  $2^{3} \cdot 7 \cdot 37^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-11796649994465343223552 $  =  $-1 \cdot 2^{8} \cdot 7 \cdot 37^{12} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{2065624967846736}{17960084863} \)  =  $-1 \cdot 2^{4} \cdot 3^{3} \cdot 7^{-1} \cdot 17^{3} \cdot 37^{-6} \cdot 991^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.0597726274394206611776785171\dots$
Stable Faltings height: $0.79221555074401156604880926728\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $115.07723816905286890066883028\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.034428073775743201630110289374\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 8 $  = $ 2\cdot1\cdot2^{2} $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L'(E,1) $ ≈ $ 7.9237752911858473667669194800125481816 $

Modular invariants

Modular form 76664.2.a.e

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q + 4q^{5} - q^{7} - 3q^{9} - 4q^{11} + 4q^{13} - 2q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 6697728
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

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

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_1^{*}$ Additive 1 3 8 0
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$37$ $4$ $I_6^{*}$ Additive 1 2 12 6

Galois representations

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

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
$2$ 2B 2.3.0.1

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add ss ordinary nonsplit ordinary ordinary ss ordinary ss ordinary ordinary add ordinary ordinary ordinary
$\lambda$-invariant(s) - 3,1 1 1 1 1 1,1 3 1,1 1 1 - 1 1 1
$\mu$-invariant(s) - 0,0 0 0 0 0 0,0 0 0,0 0 0 - 0 0 0

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

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 76664.e consists of 2 curves linked by isogenies of degree 2.

Growth of torsion in number fields

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-7}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$4$ 4.2.153328.1 \(\Z/4\Z\) Not in database
$8$ 8.0.3527890679824.3 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.1151964303616.1 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) 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.