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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -696, -7108])

gp: E = ellinit([0, 0, 0, -696, -7108])

magma: E := EllipticCurve([0, 0, 0, -696, -7108]);

$$y^2=x^3-696x-7108$$ ## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(38, 146\right)$$ $$\hat{h}(P)$$ ≈ $3.2818556446927469769959288608$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(38,\pm 146)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$1584$$ = $$2^{4} \cdot 3^{2} \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-248396544$$ = $$-1 \cdot 2^{8} \cdot 3^{6} \cdot 11^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{199794688}{1331}$$ = $$-1 \cdot 2^{13} \cdot 11^{-3} \cdot 29^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.44631846287594188359176588999\dots$$ Stable Faltings height: $$-0.56508580183140983505067814278\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$3.2818556446927469769959288608\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.46456275073365827636813923138\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$2$$  = $$2\cdot1\cdot1$$ 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);

$$q + 3q^{5} - 2q^{7} - q^{11} - 4q^{13} - 6q^{17} - 8q^{19} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 720 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$3.0492557716184919925234891235028071010$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_0^{*}$$ Additive -1 4 8 0
$$3$$ $$1$$ $$I_0^{*}$$ Additive -1 2 6 0
$$11$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3

## 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

## $p$-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.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add add ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ss ordinary ordinary ss ordinary ss - - 1 1 1 1 1 1 1 1,3 1 1 1,1 1 1,1 - - 0 0 0 0 0 0 0 0,0 0 0 0,0 0 0,0

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 1584.p 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 $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/3\Z$$ 2.0.4.1-39204.1-c1 $3$ 3.1.44.1 $$\Z/2\Z$$ Not in database $6$ 6.0.21296.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.559872.1 $$\Z/3\Z$$ Not in database $6$ 6.0.30976.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.313456656384.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ 12.0.116101021696.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.133839634067224124629835940102144.2 $$\Z/9\Z$$ Not in database $18$ 18.2.310901169707347479625728.1 $$\Z/6\Z$$ 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.