# Properties

 Label 9240.bi3 Conductor $9240$ Discriminant $2.324\times 10^{16}$ j-invariant $$\frac{143279368983686884}{22699269140625}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -109880, -11983872])

gp: E = ellinit([0, 1, 0, -109880, -11983872])

magma: E := EllipticCurve([0, 1, 0, -109880, -11983872]);

## Simplified equation

 $$y^2=x^3+x^2-109880x-11983872$$ y^2=x^3+x^2-109880x-11983872 (homogenize, simplify) $$y^2z=x^3+x^2z-109880xz^2-11983872z^3$$ y^2z=x^3+x^2z-109880xz^2-11983872z^3 (dehomogenize, simplify) $$y^2=x^3-8900307x-8709541794$$ y^2=x^3-8900307x-8709541794 (homogenize, minimize)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-224, 1200\right)$$ (-224, 1200) $\hat{h}(P)$ ≈ $1.3180786386008951706251567094$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-128, 0\right)$$, $$\left(376, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-249, 0\right)$$, $$(-224,\pm 1200)$$, $$(-149,\pm 1050)$$, $$\left(-128, 0\right)$$, $$\left(376, 0\right)$$, $$(1272,\pm 43680)$$, $$(2776,\pm 145200)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$9240$$ = $2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $23244051600000000$ = $2^{10} \cdot 3^{4} \cdot 5^{8} \cdot 7^{2} \cdot 11^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{143279368983686884}{22699269140625}$$ = $2^{2} \cdot 3^{-4} \cdot 5^{-8} \cdot 7^{-2} \cdot 11^{-4} \cdot 13^{3} \cdot 25357^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.8630720613925436953460048409\dots$ Stable Faltings height: $1.2854494109259226041649780730\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.3180786386008951706251567094\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.26502578931397619766087881154\dots$ 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: $256$  = $2\cdot2^{2}\cdot2^{3}\cdot2\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ 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: $L'(E,1)$ ≈ $5.5891973051694946858745358534$

## Modular invariants

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);

$$q + q^{3} + q^{5} + q^{7} + q^{9} - q^{11} - 2 q^{13} + q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 65536 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive -1 3 10 0
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$5$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$11$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

## 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
$2$ 2Cs 8.48.0.32

## $p$-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 split split split nonsplit ord ord ord ss ord ss ord ord ord ss - 2 6 2 1 1 3 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 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 and 4.
Its isogeny class 9240.bi consists of 6 curves linked by isogenies of degrees dividing 8.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-14})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{14})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{14})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{11})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.2303789694976.8 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.4.159879637499904.17 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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.