# Properties

 Label 152880ca3 Conductor 152880 Discriminant 10550844480383650390671360 j-invariant $$\frac{443915739051786565201}{21894701746029840}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -124597216, -512038109836]); // or

magma: E := EllipticCurve("152880ca3");

sage: E = EllipticCurve([0, 1, 0, -124597216, -512038109836]) # or

sage: E = EllipticCurve("152880ca3")

gp: E = ellinit([0, 1, 0, -124597216, -512038109836]) \\ or

gp: E = ellinit("152880ca3")

$$y^2 = x^{3} + x^{2} - 124597216 x - 512038109836$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-\frac{98300068}{17161}, \frac{263658066042}{2248091}\right)$$ $$\left(-5318, 14112\right)$$ $$\hat{h}(P)$$ ≈ 15.3244572067 4.72602666062

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(12826, 0\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$(-5318,\pm 14112)$$, $$\left(12826, 0\right)$$, $$(74330,\pm 20022528)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$152880$$ = $$2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$10550844480383650390671360$$ = $$2^{16} \cdot 3^{2} \cdot 5 \cdot 7^{18} \cdot 13^{3}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{443915739051786565201}{21894701746029840}$$ = $$2^{-4} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-12} \cdot 11^{3} \cdot 13^{-3} \cdot 37^{3} \cdot 18743^{3}$$ 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: $$40.0451876172$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.045326251014$$ 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: $$32$$  = $$2^{2}\cdot2\cdot1\cdot2^{2}\cdot1$$ 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 152880.2.a.ey

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 31850496 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$14.5207858067$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_8^{*}$$ Additive -1 4 16 4
$$3$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$4$$ $$I_12^{*}$$ Additive -1 2 18 12
$$13$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3

## 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 X13c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.

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
$$3$$ 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, 3, 4, 6 and 12.
Its isogeny class 152880ca consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $$\Q(\sqrt{455})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{7})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-21})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{65})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 $$\Q(\sqrt{-21}, \sqrt{65})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{7})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-21}, \sqrt{-195})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{7}, \sqrt{65})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
6 6.2.3240609120000.29 $$\Z/12\Z$$ 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.