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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -57408, -3362688]) # or

sage: E = EllipticCurve("224400ev2")

gp: E = ellinit([0, -1, 0, -57408, -3362688]) \\ or

gp: E = ellinit("224400ev2")

magma: E := EllipticCurve([0, -1, 0, -57408, -3362688]); // or

magma: E := EllipticCurve("224400ev2");

$$y^2 = x^{3} - x^{2} - 57408 x - 3362688$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{2773}{4}, \frac{136125}{8}\right)$$ $$\left(1752, -72600\right)$$ $$\hat{h}(P)$$ ≈ 3.2846301916829956 1.4230479021788145

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-63, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-198,\pm 450)$$, $$(-184,\pm 968)$$, $$(-123,\pm 1350)$$, $$(-88,\pm 1000)$$, $$\left(-63, 0\right)$$, $$(312,\pm 3000)$$, $$(466,\pm 8418)$$, $$(898,\pm 25854)$$, $$(1752,\pm 72600)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$224400$$ = $$2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$7168233600000000$$ = $$2^{13} \cdot 3^{2} \cdot 5^{8} \cdot 11^{4} \cdot 17$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{326940373369}{112003650}$$ = $$2^{-1} \cdot 3^{-2} \cdot 5^{-2} \cdot 11^{-4} \cdot 17^{-1} \cdot 83^{6}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.19729754203$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.317020748681$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$64$$  = $$2^{2}\cdot2\cdot2^{2}\cdot2\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

#### Modular form 224400.2.a.bi

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{3} - 2q^{7} + q^{9} - q^{11} - q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1179648 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$11.1454225896$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_5^{*}$$ Additive -1 4 13 1
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$4$$ $$I_2^{*}$$ Additive 1 2 8 2
$$11$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 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

## $p$-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.
Its isogeny class 224400ev consists of 2 curves linked by isogenies of degree 2.

## 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 $$\Q(\sqrt{34})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 4.0.30600.2 $$\Z/4\Z$$ 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.