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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -37, -78])

gp: E = ellinit([1, -1, 0, -37, -78])

magma: E := EllipticCurve([1, -1, 0, -37, -78]);

$$y^2+xy=x^3-x^2-37x-78$$ ## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{13}{4}, \frac{13}{8}\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E); ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$49$$ = $$7^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$343$$ = $$7^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$16581375$$ = $$3^{3} \cdot 5^{3} \cdot 17^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-7}]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $$-0.45256479877564976241569564900\dots$$ Stable Faltings height: $$-0.93904233603947808869203383486\dots$$

## 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: $$1.9333117056168115467330768390\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$2$$  = $$2$$ 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

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$0.96665585280840577336653841951483552682$$

## Local data

This elliptic curve is not semistable. There is only one prime 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$$(j)_{-}$$
$$7$$ $$2$$ $$III$$ Additive -1 2 3 0

## Galois representations

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 for all primes $$p < 1000$$ except those listed.

prime Image of Galois representation
$$7$$ B.1.5

For all other primes $$p$$, the image is a Borel subgroup if $$p=2$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -7 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -7 }{p}\right)=-1$$.

## $p$-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) $\mu$-invariant(s) 2 3 5 7 ordinary ss ss add ? 0,0 0,0 - ? 0,0 0,0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

An entry ? indicates that the invariants have not yet been computed.

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 $$d=$$ 2, 7 and 14.
Its isogeny class 49.a consists of 4 curves linked by isogenies of degrees dividing 14.

## 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 $K$ $2$ $$\Q(\sqrt{7})$$ $$\Z/2\Z \times \Z/2\Z$$ 2.2.28.1-49.1-a6 $4$ 4.0.1372.1 $$\Z/4\Z$$ Not in database $6$ $$\Q(\zeta_{7})$$ $$\Z/14\Z$$ Not in database $8$ 8.4.7710244864.1 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.30118144.2 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.30118144.1 $$\Z/8\Z$$ Not in database $8$ 8.0.481890304.1 $$\Z/8\Z$$ Not in database $8$ 8.2.257298363.1 $$\Z/6\Z$$ Not in database $12$ $$\Q(\zeta_{28})$$ $$\Z/2\Z \times \Z/14\Z$$ Not in database $12$ 12.0.126548911552.1 $$\Z/28\Z$$ Not in database $16$ 16.0.59447875862838378496.4 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ 16.0.232218265089212416.1 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ 16.0.66202447602479769.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $20$ 20.0.11194501700250570391613.1 $$\Z/22\Z$$ Not in database $21$ 21.3.3219905755813179726837607.1 $$\Z/14\Z$$ 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.