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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -712, -7402])

gp: E = ellinit([1, 0, 1, -712, -7402])

magma: E := EllipticCurve([1, 0, 1, -712, -7402]);

$$y^2+xy+y=x^3-712x-7402$$ trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E); ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$294$$ = $$2 \cdot 3 \cdot 7^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-236027904$$ = $$-1 \cdot 2^{15} \cdot 3 \cdot 7^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{16591834777}{98304}$$ = $$-1 \cdot 2^{-15} \cdot 3^{-1} \cdot 7^{2} \cdot 17^{3} \cdot 41^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.44691877325059208602663797667\dots$$ Stable Faltings height: $$-0.20171794310117901567514627114\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.46202858128714067109104564478\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$3$$  = $$1\cdot1\cdot3$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} - q^{8} + q^{9} - 3q^{10} + 3q^{11} + q^{12} - 4q^{13} + 3q^{15} + q^{16} - q^{18} - 4q^{19} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 180 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$1.3860857438614220132731369343339130416$$

## Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{15}$$ Non-split multiplicative 1 1 15 15
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$7$$ $$3$$ $$IV$$ Additive 1 2 4 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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
$$3$$ B.1.2

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 7 nonsplit split add 1 1 - 0 1 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 294.d consists of 2 curves linked by isogenies of degree 3.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ 2.0.3.1-28812.3-g1 $3$ 3.1.1176.1 $$\Z/2\Z$$ Not in database $3$ 3.1.11907.2 $$\Z/3\Z$$ Not in database $6$ 6.0.33191424.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.425329947.5 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.4148928.1 $$\Z/6\Z$$ Not in database $9$ 9.1.2592974683611648.12 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.1101670627147776.5 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.35018321405473573162444173312.7 $$\Z/9\Z$$ Not in database $18$ 18.0.20170553129552778141567843827712.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database $18$ 18.0.10327323202331022408482736039788544.1 $$\Z/2\Z \times \Z/6\Z$$ 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.