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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -7364036, -7692307440]) # or

sage: E = EllipticCurve("1110n2")

gp: E = ellinit([1, 0, 0, -7364036, -7692307440]) \\ or

gp: E = ellinit("1110n2")

magma: E := EllipticCurve([1, 0, 0, -7364036, -7692307440]); // or

magma: E := EllipticCurve("1110n2");

$$y^2 + x y = x^{3} - 7364036 x - 7692307440$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$1110$$ = $$2 \cdot 3 \cdot 5 \cdot 37$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-1823508000000000$$ = $$-1 \cdot 2^{11} \cdot 3^{2} \cdot 5^{9} \cdot 37^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{44164307457093068844199489}{1823508000000000}$$ = $$-1 \cdot 2^{-11} \cdot 3^{-2} \cdot 5^{-9} \cdot 7^{3} \cdot 23^{3} \cdot 37^{-3} \cdot 79^{3} \cdot 27791^{3}$$ 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); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.045823745249$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$66$$  = $$11\cdot2\cdot1\cdot3$$ 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);

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - q^{10} + 3q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} + 3q^{17} + q^{18} + 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 23760 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$3.02436718643$$

## Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$2$$ $$11$$ $$I_{11}$$ Split multiplicative -1 1 11 11
$$3$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$5$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$37$$ $$3$$ $$I_{3}$$ 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.1.2

## $p$-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 37 split split nonsplit ordinary ordinary split 11 1 0 0 2 1 0 1 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 13$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 1110n consists of 2 curves linked by isogenies of degree 3.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ Not in database
$3$ 3.1.972.2 $$\Z/3\Z$$ Not in database
$3$ 3.1.1480.1 $$\Z/2\Z$$ Not in database
$6$ 6.0.2834352.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database
$6$ 6.0.59140800.1 $$\Z/6\Z$$ Not in database
$6$ 6.0.3241792000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.