# Properties

 Label 2450y5 Conductor 2450 Discriminant -3373232128000000 j-invariant $$-\frac{548347731625}{1835008}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -208888, 36835392]) # or

sage: E = EllipticCurve("2450y5")

gp: E = ellinit([1, 0, 0, -208888, 36835392]) \\ or

gp: E = ellinit("2450y5")

magma: E := EllipticCurve([1, 0, 0, -208888, 36835392]); // or

magma: E := EllipticCurve("2450y5");

$$y^2 + x y = x^{3} - 208888 x + 36835392$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(368, -3320\right)$$ $$\hat{h}(P)$$ ≈ 0.20847208840751474

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-528, 264\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-528, 264\right)$$, $$\left(-272, 8712\right)$$, $$\left(-272, -8440\right)$$, $$\left(-24, 6480\right)$$, $$\left(-24, -6456\right)$$, $$\left(172, 2364\right)$$, $$\left(172, -2536\right)$$, $$\left(256, 264\right)$$, $$\left(256, -520\right)$$, $$\left(272, 264\right)$$, $$\left(272, -536\right)$$, $$\left(368, 2952\right)$$, $$\left(368, -3320\right)$$, $$\left(1922, 81114\right)$$, $$\left(1922, -83036\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2450$$ = $$2 \cdot 5^{2} \cdot 7^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-3373232128000000$$ = $$-1 \cdot 2^{18} \cdot 5^{6} \cdot 7^{7}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{548347731625}{1835008}$$ = $$-1 \cdot 2^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 1637^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.208472088408$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.448097824329$$ 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: $$144$$  = $$( 2 \cdot 3^{2} )\cdot2\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form2450.2.a.t

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 20736 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

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);

$$L'(E,1)$$ ≈ $$3.36297201296$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$18$$ $$I_{18}$$ Split multiplicative -1 1 18 18
$$5$$ $$2$$ $$I_0^{*}$$ Additive 1 2 6 0
$$7$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1

## 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 X16.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ and has index 6.

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 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.

## 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 split ordinary add add ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 4 5 - - 1,1 3 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

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, 6, 9 and 18.
Its isogeny class 2450y consists of 6 curves linked by isogenies of degrees dividing 18.

## 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 $$\Q(\sqrt{-7})$$ $$\Z/2\Z \times \Z/2\Z$$ 2.0.7.1-17500.2-f1
$$\Q(\sqrt{105})$$ $$\Z/6\Z$$ 2.2.105.1-28.1-b1
4 $$\Q(\sqrt{-7}, \sqrt{-15})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
4.2.11200.2 $$\Z/4\Z$$ Not in database
6 6.0.1531537875.1 $$\Z/6\Z$$ Not in database
6.6.41351522625.1 $$\Z/18\Z$$ 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.