# Properties

 Label 960.p3 Conductor $960$ Discriminant $2.359\times 10^{12}$ j-invariant $$\frac{4102915888729}{9000000}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-21345x+1190943$$ y^2=x^3+x^2-21345x+1190943 (homogenize, simplify) $$y^2z=x^3+x^2z-21345xz^2+1190943z^3$$ y^2z=x^3+x^2z-21345xz^2+1190943z^3 (dehomogenize, simplify) $$y^2=x^3-1728972x+873384336$$ y^2=x^3-1728972x+873384336 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -21345, 1190943])

gp: E = ellinit([0, 1, 0, -21345, 1190943])

magma: E := EllipticCurve([0, 1, 0, -21345, 1190943]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(81, 0\right)$$, $$\left(87, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-169, 0\right)$$, $$\left(81, 0\right)$$, $$\left(87, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$960$$ = $2^{6} \cdot 3 \cdot 5$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $2359296000000$ = $2^{24} \cdot 3^{2} \cdot 5^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4102915888729}{9000000}$$ = $2^{-6} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{3} \cdot 2287^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.2573131657762969897064744430\dots$ Stable Faltings height: $0.21759239493637902558062626081\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.81910691718318947792851914759\dots$ 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: $48$  = $2^{2}\cdot2\cdot( 2 \cdot 3 )$ 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: $L(E,1)$ ≈ $2.4573207515495684337855574428$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2304 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $4$ $I_{14}^{*}$ Additive -1 6 24 6
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$5$ $6$ $I_{6}$ Split multiplicative -1 1 6 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
$2$ 2Cs 4.12.0.1
$3$ 3B 3.4.0.1

The image of the adelic Galois representation has level $120$, index $384$, and genus $5$.

## $p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 5 add split split - 3 1 - 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 960.p consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $K$ $2$ $$\Q(\sqrt{6})$$ $$\Z/2\Z \oplus \Z/12\Z$$ 2.2.24.1-150.1-b9 $4$ $$\Q(i, \sqrt{10})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-6}, \sqrt{10})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $6$ 6.0.30233088.2 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.3317760000.5 $$\Z/4\Z \oplus \Z/12\Z$$ Not in database $8$ 8.4.76441190400.17 $$\Z/2\Z \oplus \Z/24\Z$$ Not in database $12$ 12.0.8226356490141696.17 $$\Z/6\Z \oplus \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $18$ 18.6.1295354998363672018944000000000000.6 $$\Z/2\Z \oplus \Z/36\Z$$ 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.