# Properties

 Label 390390bf3 Conductor 390390 Discriminant 272695202158128750 j-invariant $$\frac{3971101377248209009}{56495958750}$$ 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, 1, 0, -5575482, 5064858414]); // or

magma: E := EllipticCurve("390390bf3");

sage: E = EllipticCurve([1, 1, 0, -5575482, 5064858414]) # or

sage: E = EllipticCurve("390390bf3")

gp: E = ellinit([1, 1, 0, -5575482, 5064858414]) \\ or

gp: E = ellinit("390390bf3")

$$y^2 + x y = x^{3} + x^{2} - 5575482 x + 5064858414$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-957, 98076\right)$$ $$\left(1383, -1959\right)$$ $$\hat{h}(P)$$ ≈ 0.470050334507 1.16528385285

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(\frac{5467}{4}, -\frac{5467}{8}\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-2607, 44781\right)$$, $$\left(-2607, -42174\right)$$, $$\left(-957, 98076\right)$$, $$\left(-957, -97119\right)$$, $$\left(-385, 84777\right)$$, $$\left(-385, -84392\right)$$, $$\left(1045, 18997\right)$$, $$\left(1045, -20042\right)$$, $$\left(1353, -99\right)$$, $$\left(1353, -1254\right)$$, $$\left(1383, 576\right)$$, $$\left(1383, -1959\right)$$, $$\left(1409, 2253\right)$$, $$\left(1409, -3662\right)$$, $$\left(1523, 9676\right)$$, $$\left(1523, -11199\right)$$, $$\left(2123, 51876\right)$$, $$\left(2123, -53999\right)$$, $$\left(3333, 151701\right)$$, $$\left(3333, -155034\right)$$, $$\left(10873, 1103626\right)$$, $$\left(10873, -1114499\right)$$, $$\left(23613, 3599301\right)$$, $$\left(23613, -3622914\right)$$, $$\left(102783, 32892201\right)$$, $$\left(102783, -32994984\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$390390$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$272695202158128750$$ = $$2 \cdot 3^{2} \cdot 5^{4} \cdot 7^{3} \cdot 11^{4} \cdot 13^{6}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{3971101377248209009}{56495958750}$$ = $$2^{-1} \cdot 3^{-2} \cdot 5^{-4} \cdot 7^{-3} \cdot 11^{-4} \cdot 13^{3} \cdot 181^{3} \cdot 673^{3}$$ 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: $$0.427951699353$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.282528744781$$ 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: $$384$$  = $$1\cdot2\cdot2^{2}\cdot3\cdot2^{2}\cdot2^{2}$$ 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 390390.2.a.bf

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)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + q^{11} - q^{12} - q^{14} - q^{15} + q^{16} + 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 14155776 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### 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])

$$L^{(2)}(E,1)/2!$$ ≈ $$11.6072310187$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$11$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$13$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0

## 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 X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 390390bf consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $$\Q(\sqrt{14})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{91})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{26})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\sqrt{14}, \sqrt{26})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.