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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -156, -722]) # or

sage: E = EllipticCurve("2646.a2")

gp: E = ellinit([1, -1, 0, -156, -722]) \\ or

gp: E = ellinit("2646.a2")

magma: E := EllipticCurve([1, -1, 0, -156, -722]); // or

magma: E := EllipticCurve("2646.a2");

$$y^2+xy=x^3-x^2-156x-722$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$2646$$ = $$2 \cdot 3^{3} \cdot 7^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-6353046$$ = $$-1 \cdot 2 \cdot 3^{3} \cdot 7^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{132651}{2}$$ = $$-1 \cdot 2^{-1} \cdot 3^{3} \cdot 17^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## 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: $$0.67463491165153561743977262970$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$1$$  = $$1\cdot1\cdot1$$ 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);

$$q - q^{2} + q^{4} - 3q^{5} - q^{8} + 3q^{10} - 3q^{11} + 4q^{13} + q^{16} - 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 756 $$\Gamma_0(N)$$-optimal: yes 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);

$$L(E,1)$$ ≈ $$0.67463491165153561743977262970543442670$$

## Local data

This elliptic curve is 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$3$$ $$1$$ $$II$$ Additive 1 3 3 0
$$7$$ $$1$$ $$I_0^{*}$$ Additive -1 2 6 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

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 11 13 17 19 23 29 31 37 41 43 47 nonsplit add ordinary add ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 1 - 0 - 0 0 0,0 0 0 0 0 0 0 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 and 9.
Its isogeny class 2646.a consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/3\Z$$ 2.0.7.1-2916.2-a1 $3$ 3.1.216.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1119744.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.12002256.1 $$\Z/3\Z$$ Not in database $6$ 6.0.2250423.1 $$\Z/9\Z$$ Not in database $6$ 6.0.16003008.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.144054149089536.4 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.2.14701569336408730423883268096.1 $$\Z/9\Z$$ Not in database $18$ 18.2.2294529340113174813908926464.1 $$\Z/6\Z$$ Not in database $18$ 18.0.2178010272060552655390113792.1 $$\Z/18\Z$$ 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.