# Properties

 Label 86190.s1 Conductor 86190 Discriminant -191352323830876923312000 j-invariant $$\frac{414491631408272360789951}{1132262271188620848000}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, 8588018, -18680853836]); // or

magma: E := EllipticCurve("86190p1");

sage: E = EllipticCurve([1, 1, 0, 8588018, -18680853836]) # or

sage: E = EllipticCurve("86190p1")

gp: E = ellinit([1, 1, 0, 8588018, -18680853836]) \\ or

gp: E = ellinit("86190p1")

$$y^2 + x y = x^{3} + x^{2} + 8588018 x - 18680853836$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(\frac{25147}{9}, \frac{4410202}{27}\right)$$ $$\hat{h}(P)$$ ≈ 7.5265341432

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$86190$$ = $$2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-191352323830876923312000$$ = $$-1 \cdot 2^{7} \cdot 3^{25} \cdot 5^{3} \cdot 13^{2} \cdot 17^{4}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{414491631408272360789951}{1132262271188620848000}$$ = $$2^{-7} \cdot 3^{-25} \cdot 5^{-3} \cdot 13 \cdot 17^{-4} \cdot 1951^{3} \cdot 16253^{3}$$ 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: $$7.5265341432$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0518314936118$$ 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: $$12$$  = $$1\cdot1\cdot3\cdot1\cdot2^{2}$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 86190.2.a.s

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 9676800 $$\Gamma_0(N)$$-optimal: yes 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])

$$L'(E,1)$$ ≈ $$4.68133807634$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{7}$$ Non-split multiplicative 1 1 7 7
$$3$$ $$1$$ $$I_{25}$$ Non-split multiplicative 1 1 25 25
$$5$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$13$$ $$1$$ $$II$$ Additive 1 2 2 0
$$17$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4

## 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 $$p$$ .

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

## 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 split ordinary ordinary add split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 6 1 2 5 1 - 2 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 no rational isogenies. Its isogeny class 86190.s consists of this curve only.

## 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
3 3.1.20280.1 $$\Z/2\Z$$ Not in database
6 6.0.49353408000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.