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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 1522, 95915])

gp: E = ellinit([1, 1, 1, 1522, 95915])

magma: E := EllipticCurve([1, 1, 1, 1522, 95915]);

$$y^2+xy+y=x^3+x^2+1522x+95915$$ ## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(5, 319\right)$$ $$\hat{h}(P)$$ ≈ $0.93136729335260952445799882365$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{141}{4}, \frac{137}{8}\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-5, 299\right)$$, $$\left(-5, -295\right)$$, $$\left(5, 319\right)$$, $$\left(5, -325\right)$$, $$\left(93, 979\right)$$, $$\left(93, -1073\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$348726$$ = $$2 \cdot 3 \cdot 7 \cdot 19^{2} \cdot 23$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-4147535018592$$ = $$-1 \cdot 2^{5} \cdot 3^{6} \cdot 7^{2} \cdot 19^{3} \cdot 23^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{56844576125}{604685088}$$ = $$2^{-5} \cdot 3^{-6} \cdot 5^{3} \cdot 7^{-2} \cdot 23^{-2} \cdot 769^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.1021703265913437280194647914\dots$$ Stable Faltings height: $$0.36606058179973361301720793343\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.93136729335260952445799882365\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.57408892882409038547247851351\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$80$$  = $$5\cdot2\cdot2\cdot2\cdot2$$ 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 348726.2.a.by

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - q^{12} + 2q^{13} + q^{14} + q^{16} - 6q^{17} + q^{18} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 614400 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$10.693753035651839193881865675095467896$$

## Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

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$$(j)_{-}$$
$$2$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$3$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$19$$ $$2$$ $$III$$ Additive 1 2 3 0
$$23$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 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 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(5,20) 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 348726.by consists of 2 curves linked by isogenies of degree 2.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-38})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.2.51204136032.2 $$\Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.