Properties

 Label 286110.ch1 Conductor 286110 Discriminant -2601519967414465218000 j-invariant $$-\frac{4368317413923}{5475734000}$$ 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, -2657409, -2966187187]); // or

magma: E := EllipticCurve("286110ch1");

sage: E = EllipticCurve([1, -1, 0, -2657409, -2966187187]) # or

sage: E = EllipticCurve("286110ch1")

gp: E = ellinit([1, -1, 0, -2657409, -2966187187]) \\ or

gp: E = ellinit("286110ch1")

$$y^2 + x y = x^{3} - x^{2} - 2657409 x - 2966187187$$

Mordell-Weil group structure

$$\Z$$

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(9142, 853759\right)$$ $$\hat{h}(P)$$ ≈ 0.356501649659

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(2597, 86124\right)$$, $$\left(2597, -88721\right)$$, $$\left(4907, 317124\right)$$, $$\left(4907, -322031\right)$$, $$\left(9142, 853759\right)$$, $$\left(9142, -862901\right)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$286110$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-2601519967414465218000$$ = $$-1 \cdot 2^{4} \cdot 3^{9} \cdot 5^{3} \cdot 11^{5} \cdot 17^{7}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{4368317413923}{5475734000}$$ = $$-1 \cdot 2^{-4} \cdot 3^{3} \cdot 5^{-3} \cdot 11^{-5} \cdot 17^{-1} \cdot 5449^{3}$$ 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: $$0.356501649659$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0564524857396$$ 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: $$240$$  = $$2\cdot2\cdot3\cdot5\cdot2^{2}$$ 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 286110.2.a.ch

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)

$$q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + q^{11} + q^{13} + q^{14} + q^{16} - 7q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 16588800 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$4.83009703044$$

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$2$$ $$III^{*}$$ Additive 1 2 9 0
$$5$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$11$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$17$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 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 $$p$$ .

$p$-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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has no rational isogenies. Its isogeny class 286110.ch 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 3.1.11220.1 $$\Z/2\Z$$ Not in database
6 6.0.1412467848000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.