# Properties

 Label 2415h2 Conductor 2415 Discriminant 4251692025 j-invariant $$\frac{7347774183121}{4251692025}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -405, 0]); // or

magma: E := EllipticCurve("2415h2");

sage: E = EllipticCurve([1, 0, 0, -405, 0]) # or

sage: E = EllipticCurve("2415h2")

gp: E = ellinit([1, 0, 0, -405, 0]) \\ or

gp: E = ellinit("2415h2")

$$y^2 + x y = x^{3} - 405 x$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-15, 60\right)$$ $$\hat{h}(P)$$ ≈ 0.537930241608

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(0, 0\right)$$, $$\left(20, -10\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-15, 60\right)$$, $$\left(-15, -45\right)$$, $$\left(-3, 36\right)$$, $$\left(-3, -33\right)$$, $$\left(0, 0\right)$$, $$\left(20, -10\right)$$, $$\left(27, 81\right)$$, $$\left(27, -108\right)$$, $$\left(36, 162\right)$$, $$\left(36, -198\right)$$, $$\left(135, 1485\right)$$, $$\left(135, -1620\right)$$, $$\left(867, 25092\right)$$, $$\left(867, -25959\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$2415$$ = $$3 \cdot 5 \cdot 7 \cdot 23$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$4251692025$$ = $$3^{8} \cdot 5^{2} \cdot 7^{2} \cdot 23^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{7347774183121}{4251692025}$$ = $$3^{-8} \cdot 5^{-2} \cdot 7^{-2} \cdot 23^{-2} \cdot 19441^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.537930241608$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$1.17063697465$$ 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: $$64$$  = $$2^{3}\cdot2\cdot2\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$4$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form2415.2.a.c

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)

$$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} - q^{10} - q^{12} + 2q^{13} + q^{14} + q^{15} - q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1024 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$2.51888412243$$

## Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$3$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$23$$ $$2$$ $$I_{2}$$ Non-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 X8d.

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

## $p$-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.

## 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 ordinary split split nonsplit ss ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ss 6 2 2 1 1,1 1 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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 2415h 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-5})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
4 $$\Q(\sqrt{5}, \sqrt{161})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(i, \sqrt{161})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.