# Properties

 Label 227430.ec5 Conductor $227430$ Discriminant $2.964\times 10^{17}$ j-invariant $$\frac{37966934881}{8643600}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -227498, -32471719]) # or

sage: E = EllipticCurve("227430.ec5")

gp: E = ellinit([1, -1, 1, -227498, -32471719]) \\ or

gp: E = ellinit("227430.ec5")

magma: E := EllipticCurve([1, -1, 1, -227498, -32471719]); // or

magma: E := EllipticCurve("227430.ec5");

$$y^2+xy+y=x^3-x^2-227498x-32471719$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(651, 9421\right)$$ $$\hat{h}(P)$$ ≈ $1.7039098448886747399900091738$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-375, 187\right)$$, $$\left(537, -269\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-375, 187\right)$$, $$\left(-345, 2377\right)$$, $$\left(-345, -2033\right)$$, $$\left(-185, 1897\right)$$, $$\left(-185, -1713\right)$$, $$\left(537, -269\right)$$, $$\left(651, 9421\right)$$, $$\left(651, -10073\right)$$, $$\left(1561, 57651\right)$$, $$\left(1561, -59213\right)$$, $$\left(6123, 474541\right)$$, $$\left(6123, -480665\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  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: $$296444771441456400$$ = $$2^{4} \cdot 3^{8} \cdot 5^{2} \cdot 7^{4} \cdot 19^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{37966934881}{8643600}$$ = $$2^{-4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-4} \cdot 3361^{3}$$ 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); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.7039098448886747399900091738$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.22217473104297402435978540471$$ 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: $$512$$  = $$2^{2}\cdot2^{2}\cdot2\cdot2^{2}\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 227430.2.a.ec

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);

$$q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} - 4q^{11} + 2q^{13} + q^{14} + q^{16} - 2q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3538944 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$12.114102768307740751516777907585357973$$

## Local data

This elliptic curve is semistable. There are 5 primes of bad reduction:

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$19$$ $$4$$ $$I_0^{*}$$ 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 X101.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 6 \\ 4 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 7 \end{array}\right)$ and has index 24.

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
$$2$$ 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.

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 227430.ec consists of 3 curves linked by isogenies of degrees dividing 8.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{57})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-15}, \sqrt{-57})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{15}, \sqrt{-57})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.1688960160000.7 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.380016036000000.60 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.103812949610496.196 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.