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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -42916650, 42585182836]) # or

sage: E = EllipticCurve("227430fz2")

gp: E = ellinit([1, -1, 0, -42916650, 42585182836]) \\ or

gp: E = ellinit("227430fz2")

magma: E := EllipticCurve([1, -1, 0, -42916650, 42585182836]); // or

magma: E := EllipticCurve("227430fz2");

$$y^2 + x y = x^{3} - x^{2} - 42916650 x + 42585182836$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-147, 221186\right)$$ $$\hat{h}(P)$$ ≈ $1.1169296683220844$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{4067}{4}, -\frac{4067}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-3577, 389501\right)$$, $$\left(-3577, -385924\right)$$, $$\left(-147, 221186\right)$$, $$\left(-147, -221039\right)$$, $$\left(35231, 6483092\right)$$, $$\left(35231, -6518323\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$227430$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$4275879135099141613478400$$ = $$2^{9} \cdot 3^{3} \cdot 5^{2} \cdot 7^{12} \cdot 19^{7}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{6882017790203934867}{3366201047283200}$$ = $$2^{-9} \cdot 3^{12} \cdot 5^{-2} \cdot 7^{-12} \cdot 19^{-1} \cdot 23^{3} \cdot 1021^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential 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: $$1.11692966832$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.069116819443$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$192$$  = $$1\cdot2\cdot2\cdot( 2^{2} \cdot 3 )\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 227430.2.a.z

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} - 2q^{13} - q^{14} + q^{16} - 6q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 44789760 $$\Gamma_0(N)$$-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/factorial(ar)

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L'(E,1)$$ ≈ $$3.70553405836$$

## Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$19$$ $$4$$ $$I_1^{*}$$ 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 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.

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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 227430fz consists of 4 curves linked by isogenies of degrees dividing 6.

## 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
$2$ $$\Q(\sqrt{-19})$$ $$\Z/6\Z$$ Not in database
$2$ $$\Q(\sqrt{114})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$4$ $$\Q(\sqrt{-6}, \sqrt{-19})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$4$ 4.0.410400.2 $$\Z/4\Z$$ Not in database
$6$ 6.2.3384517820625.1 $$\Z/6\Z$$ 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.