# Properties

 Label 86190.d4 Conductor 86190 Discriminant 326543330736090000 j-invariant $$\frac{330240275458561}{67652010000}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -243363, 37039293]) # or

sage: E = EllipticCurve("86190d3")

gp: E = ellinit([1, 1, 0, -243363, 37039293]) \\ or

gp: E = ellinit("86190d3")

magma: E := EllipticCurve([1, 1, 0, -243363, 37039293]); // or

magma: E := EllipticCurve("86190d3");

$$y^2 + x y = x^{3} + x^{2} - 243363 x + 37039293$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-502, 5997\right)$$ $$\left(-451, 7663\right)$$ $$\hat{h}(P)$$ ≈ 1.3012674937263806 2.915413710880036

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(174, -87\right)$$, $$\left(382, -191\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-502, 5997\right)$$, $$\left(-502, -5495\right)$$, $$\left(-451, 7663\right)$$, $$\left(-451, -7212\right)$$, $$\left(-294, 9273\right)$$, $$\left(-294, -8979\right)$$, $$\left(93, 3855\right)$$, $$\left(93, -3948\right)$$, $$\left(161, 1356\right)$$, $$\left(161, -1517\right)$$, $$\left(174, -87\right)$$, $$\left(382, -191\right)$$, $$\left(399, 1713\right)$$, $$\left(399, -2112\right)$$, $$\left(499, 6088\right)$$, $$\left(499, -6587\right)$$, $$\left(2983, 159337\right)$$, $$\left(2983, -162320\right)$$, $$\left(5907, 449544\right)$$, $$\left(5907, -455451\right)$$, $$\left(11874, 1286913\right)$$, $$\left(11874, -1298787\right)$$, $$\left(67774, 17609713\right)$$, $$\left(67774, -17677487\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$86190$$ = $$2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$326543330736090000$$ = $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{6} \cdot 17^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{330240275458561}{67652010000}$$ = $$2^{-4} \cdot 3^{-4} \cdot 5^{-4} \cdot 13^{6} \cdot 17^{-4} \cdot 409^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.75175254592$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.288580999813$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$128$$  = $$2\cdot2\cdot2\cdot2^{2}\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 86190.2.a.d

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

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

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1179648 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L^{(2)}(E,1)/2!$$ ≈ $$6.3528280075$$

## Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$5$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0
$$17$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4

## 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 X183.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 6 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 6 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 7 \end{array}\right)$ and has index 48.

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
$$2$$ 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 nonsplit nonsplit nonsplit ss ordinary add split ordinary ss ordinary ordinary ordinary ordinary ordinary ss 6 2 2 2,4 2 - 3 2 2,2 2 2 2 2 2 2,2 1 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 $$d=$$ 2 and 4.
Its isogeny class 86190.d consists of 8 curves linked by isogenies of degrees dividing 16.

## 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
2 $$\Q(\sqrt{-13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
4 $$\Q(i, \sqrt{13})$$ $$\Z/4\Z \times \Z/4\Z$$ 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.