# Properties

 Label 321346.b2 Conductor $$321346$$ Discriminant $$-13971531281465344$$ j-invariant $$-\frac{90389301567546411793}{13971531281465344}$$ CM no Rank $$2$$ Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -93497, 12378745]); // or
magma: E := EllipticCurve("321346b1");
sage: E = EllipticCurve([1, 0, 0, -93497, 12378745]) # or
sage: E = EllipticCurve("321346b1")
gp: E = ellinit([1, 0, 0, -93497, 12378745]) \\ or
gp: E = ellinit("321346b1")

$$y^2 + x y = x^{3} - 93497 x + 12378745$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-196, 4913\right)$$ $$\left(42, 2899\right)$$ $$\hat{h}(P)$$ ≈ 5.44035100039 1.24961710052

## Torsion generators

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

$$\left(-358, 179\right)$$

## Integral points

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

$$\left(-358, 179\right)$$, $$\left(-196, 4913\right)$$, $$\left(42, 2899\right)$$, $$\left(154, 1203\right)$$, $$\left(210, 1315\right)$$, $$\left(1210, 40275\right)$$, $$\left(67354, 17446323\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$321346$$ = $$2 \cdot 31 \cdot 71 \cdot 73$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-13971531281465344$$ = $$-1 \cdot 2^{24} \cdot 31 \cdot 71^{2} \cdot 73^{2}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{90389301567546411793}{13971531281465344}$$ = $$-1 \cdot 2^{-24} \cdot 11^{3} \cdot 19^{3} \cdot 31^{-1} \cdot 71^{-2} \cdot 73^{-2} \cdot 109^{3} \cdot 197^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$2$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1.87415264613$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$0.382603078154$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$96$$  = $$( 2^{3} \cdot 3 )\cdot1\cdot2\cdot2$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 321346.2.a.b

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

$$q + q^{2} - 2q^{3} + q^{4} + 2q^{5} - 2q^{6} + q^{8} + q^{9} + 2q^{10} - 2q^{12} - 4q^{13} - 4q^{15} + q^{16} - 4q^{17} + q^{18} - 8q^{19} + O(q^{20})$$

#### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
2260992 . This curve is $$\Gamma_0(N)$$-optimal.

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

$$L^{(2)}(E,1)/2!$$ ≈ $$17.2093577122$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$24$$ $$I_{24}$$ Split multiplicative -1 1 24 24
$$31$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$71$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$73$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

No $$p$$-adic data exists for this curve.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 321346.b consists of 2 curves linked by isogenies of degree 2.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 $$\Q(\sqrt{-31})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 4.2.2500336.1 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.