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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -45277633, 61546572863]) # or

sage: E = EllipticCurve("235200ux2")

gp: E = ellinit([0, 1, 0, -45277633, 61546572863]) \\ or

gp: E = ellinit("235200ux2")

magma: E := EllipticCurve([0, 1, 0, -45277633, 61546572863]); // or

magma: E := EllipticCurve("235200ux2");

$$y^2 = x^{3} + x^{2} - 45277633 x + 61546572863$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-1702, 365625\right)$$ $$\left(-271, 271656\right)$$ $$\hat{h}(P)$$ ≈ 3.5001809461366076 3.8105399916932567

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(5903, 0\right)$$, $$\left(1423, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-7327, 0\right)$$, $$(-6641,\pm 263424)$$, $$(-3697,\pm 422400)$$, $$(-1702,\pm 365625)$$, $$(-271,\pm 271656)$$, $$(1178,\pm 99225)$$, $$\left(1423, 0\right)$$, $$\left(5903, 0\right)$$, $$(13253,\pm 1337700)$$, $$(13967,\pm 1467648)$$, $$(161423,\pm 64800000)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$235200$$ = $$2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$4303400887296000000000000$$ = $$2^{22} \cdot 3^{6} \cdot 5^{12} \cdot 7^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{21302308926361}{8930250000}$$ = $$2^{-4} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-2} \cdot 19^{3} \cdot 1459^{3}$$ 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: $$9.82448772587$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.070319068518$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$384$$  = $$2^{2}\cdot( 2 \cdot 3 )\cdot2^{2}\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 235200.2.a.ux

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 42467328 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$16.5803718132$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_12^{*}$$ Additive -1 6 22 4
$$3$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$5$$ $$4$$ $$I_6^{*}$$ Additive 1 2 12 6
$$7$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 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 X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by  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
$$2$$ Cs
$$3$$ 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 235200ux consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $$\Q(\sqrt{70})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
4 $$\Q(\sqrt{-5}, \sqrt{14})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-14}, \sqrt{-30})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{21}, \sqrt{30})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.116169984000.9 $$\Z/2\Z \times \Z/6\Z$$ 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.