Properties

 Label 115920cy4 Conductor $115920$ Discriminant $9.051\times 10^{23}$ j-invariant $$\frac{12411881707829361287041}{303132494474220600}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -69459843, -218064841342])

gp: E = ellinit([0, 0, 0, -69459843, -218064841342])

magma: E := EllipticCurve([0, 0, 0, -69459843, -218064841342]);

$$y^2=x^3-69459843x-218064841342$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-5071, 61360\right)$$ $\hat{h}(P)$ ≈ $5.6090776788116818129906646517$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-4226, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-5071,\pm 61360)$$, $$\left(-4226, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$115920$$ = $2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $905148778380111124070400$ = $2^{15} \cdot 3^{8} \cdot 5^{2} \cdot 7^{12} \cdot 23^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{12411881707829361287041}{303132494474220600}$$ = $2^{-3} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-12} \cdot 23^{-3} \cdot 29^{3} \cdot 47^{3} \cdot 16987^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.3809385951469705606110395075\dots$ Stable Faltings height: $2.1384852702529704054961847676\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $5.6090776788116818129906646517\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.052373618374511291775508278189\dots$ 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: $64$  = $2^{2}\cdot2^{2}\cdot2\cdot2\cdot1$ 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: $L'(E,1)$ ≈ $4.7002831005291623099528477219162320826$

Modular invariants

Modular form 115920.2.a.a

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);

$$q - q^{5} - q^{7} - 6q^{11} - 4q^{13} + 6q^{17} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 23887872 $\Gamma_0(N)$-optimal: no Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $4$ $I_7^{*}$ Additive -1 4 15 3
$3$ $4$ $I_2^{*}$ Additive -1 2 8 2
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $2$ $I_{12}$ Non-split multiplicative 1 1 12 12
$23$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3

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 $\GL(2,\Z_\ell)$ 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
$3$ 3B 3.4.0.1

$p$-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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add add nonsplit nonsplit ordinary ordinary ordinary ordinary nonsplit ss ordinary ordinary ordinary ordinary ss - - 1 1 1 1 1 1 1 1,1 1 1 1 1 1,1 - - 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, 3 and 6.
Its isogeny class 115920cy consists of 4 curves linked by isogenies of degrees dividing 6.

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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{46})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/6\Z$$ Not in database $4$ 4.0.41400.3 $$\Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{46})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.2.7085880000.7 $$\Z/6\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.58027829760000.3 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.109693440000.28 $$\Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $18$ 18.0.6395111348672761268877660114828596556915609600000000.2 $$\Z/18\Z$$ 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.