Properties

Label 81120.br1
Conductor $81120$
Discriminant $151856640000$
j-invariant \( \frac{150568768}{16875} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -2305, -39025])
 
gp: E = ellinit([0, 1, 0, -2305, -39025])
 
magma: E := EllipticCurve([0, 1, 0, -2305, -39025]);
 

\(y^2=x^3+x^2-2305x-39025\)  Toggle raw display

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

$P$ =  \(\left(-25, 60\right)\)  Toggle raw display\(\left(55, 60\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.37286905124459088813510879367$$1.9916738154309061732405580203$

Torsion generators

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

\( \left(-35, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-35, 0\right) \), \((-31,\pm 60)\), \((-25,\pm 60)\), \((-22,\pm 39)\), \((55,\pm 60)\), \((65,\pm 300)\), \((95,\pm 780)\), \((173,\pm 2184)\), \((290,\pm 4875)\), \((455,\pm 9660)\), \((1135,\pm 38220)\), \((1175,\pm 40260)\), \((3847133,\pm 7545809856)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 81120 \)  =  $2^{5} \cdot 3 \cdot 5 \cdot 13^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $151856640000 $  =  $2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 13^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{150568768}{16875} \)  =  $2^{6} \cdot 3^{-3} \cdot 5^{-4} \cdot 7^{3} \cdot 19^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.87838593315852178830636221382\dots$
Stable Faltings height: $-0.45599858676680770512424176803\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $2$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.72960522628740703415041954661\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.69401533570646377179910353463\dots$
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: $ 96 $  = $ 2^{2}\cdot3\cdot2^{2}\cdot2 $
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: $1$ (rounded)
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^{(2)}(E,1)/2! $ ≈ $ 12.152573185321086220190397129579507947 $

Modular invariants

Modular form 81120.2.a.br

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);
 

\( q + q^{3} + q^{5} - 4q^{7} + q^{9} - 2q^{11} + q^{15} - 2q^{17} - 8q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 129024
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 4 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$(j)_{-}$
$2$ $4$ $I_3^{*}$ Additive 1 5 12 0
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $2$ $III$ Additive -1 2 3 0

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 split split ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - 9 3 2 4 - 2 2 2 2 2 2 2 2 2
$\mu$-invariant(s) - 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 81120.br 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
$2$ \(\Q(\sqrt{39}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$4$ 4.0.105456.1 \(\Z/4\Z\) Not in database
$8$ 8.0.1601419382784.7 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.4.922417564483584.22 \(\Z/2\Z \times \Z/4\Z\) 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.