# Properties

 Label 91035.q2 Conductor 91035 Discriminant 193999073006025 j-invariant $$\frac{47045881}{11025}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -19562, 817224]); // or

magma: E := EllipticCurve("91035bi2");

sage: E = EllipticCurve([1, -1, 1, -19562, 817224]) # or

sage: E = EllipticCurve("91035bi2")

gp: E = ellinit([1, -1, 1, -19562, 817224]) \\ or

gp: E = ellinit("91035bi2")

$$y^2 + x y + y = x^{3} - x^{2} - 19562 x + 817224$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-106, 1353\right)$$ $$\hat{h}(P)$$ ≈ 1.40081975747

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(47, -24\right)$$, $$\left(\frac{443}{4}, -\frac{447}{8}\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-157, 78\right)$$, $$\left(-106, 1353\right)$$, $$\left(-106, -1248\right)$$, $$\left(32, 456\right)$$, $$\left(32, -489\right)$$, $$\left(47, -24\right)$$, $$\left(132, 656\right)$$, $$\left(132, -789\right)$$, $$\left(243, 3098\right)$$, $$\left(243, -3342\right)$$, $$\left(914, 26853\right)$$, $$\left(914, -27768\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$91035$$ = $$3^{2} \cdot 5 \cdot 7 \cdot 17^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$193999073006025$$ = $$3^{8} \cdot 5^{2} \cdot 7^{2} \cdot 17^{6}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{47045881}{11025}$$ = $$3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 19^{6}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.40081975747$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.532518025196$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$64$$  = $$2^{2}\cdot2\cdot2\cdot2^{2}$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$4$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 91035.2.a.q

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - q^{2} - q^{4} + q^{5} - q^{7} + 3q^{8} - q^{10} - 6q^{13} + q^{14} - q^{16} - 8q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 327680 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

$$L'(E,1)$$ ≈ $$2.98384708361$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$17$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by  and has index 6.

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
$$2$$ Cs

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 ordinary add split nonsplit ss ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 11 - 2 1 1,1 1 - 1 1 1 1 1 1 1 1 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 $$d=$$ 2.
Its isogeny class 91035.q consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{105}, \sqrt{119})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-5}, \sqrt{51})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{21}, \sqrt{-51})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.