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## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, 296, 1702]); // or
magma: E := EllipticCurve("110c2");
sage: E = EllipticCurve([1, 0, 1, 296, 1702]) # or
sage: E = EllipticCurve("110c2")
gp: E = ellinit([1, 0, 1, 296, 1702]) \\ or
gp: E = ellinit("110c2")

$$y^2 + x y + y = x^{3} + 296 x + 1702$$

Trivial

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()
None

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$110$$ = $$2 \cdot 5 \cdot 11$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-2883584000$$ = $$-1 \cdot 2^{21} \cdot 5^{3} \cdot 11$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{2882081488391}{2883584000}$$ = $$2^{-21} \cdot 5^{-3} \cdot 7^{3} \cdot 11^{-1} \cdot 19^{3} \cdot 107^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$0$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$0.942057966503$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$1$$  = $$1\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form110.2.a.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 84 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar/factorial(ar)

$$L(E,1)$$ ≈ $$0.942057966503$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$I_{21}$$ Non-split multiplicative 1 1 21 21
$$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$11$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## Galois representations

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

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

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 11 nonsplit ordinary nonsplit split 2 2 0 1 0 1 0 0

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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 110c 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 $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ 2.0.3.1-12100.1-c2
3 3.1.440.1 $$\Z/2\Z$$ Not in database
3.1.3267.1 $$\Z/3\Z$$ Not in database
6 6.0.85184000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database
6.0.5227200.1 $$\Z/6\Z$$ Not in database
6.0.32019867.2 $$\Z/3\Z \times \Z/3\Z$$ 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.