Properties

 Label 302330.bo1 Conductor 302330 Discriminant -280338295433400091005998390 j-invariant $$-\frac{226953328047600468451761}{2382836194386693393110}$$ CM no Rank 0 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -62268303, 827481250957]); // or

magma: E := EllipticCurve("302330bo2");

sage: E = EllipticCurve([1, -1, 1, -62268303, 827481250957]) # or

sage: E = EllipticCurve("302330bo2")

gp: E = ellinit([1, -1, 1, -62268303, 827481250957]) \\ or

gp: E = ellinit("302330bo2")

$$y^2 + x y + y = x^{3} - x^{2} - 62268303 x + 827481250957$$

Trivial

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$302330$$ = $$2 \cdot 5 \cdot 7^{2} \cdot 617$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-280338295433400091005998390$$ = $$-1 \cdot 2 \cdot 5 \cdot 7^{7} \cdot 617^{7}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{226953328047600468451761}{2382836194386693393110}$$ = $$-1 \cdot 2^{-1} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{-1} \cdot 13^{3} \cdot 43^{3} \cdot 617^{-7} \cdot 36373^{3}$$ 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[1] Real period: $$0.0467768566801$$ 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: $$28$$  = $$1\cdot1\cdot2^{2}\cdot7$$ 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: $$9$$ (exact)

Modular invariants

Modular form 302330.2.a.bo

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 293190912 $$\Gamma_0(N)$$-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])

$$L(E,1)$$ ≈ $$11.7877678834$$

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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$617$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7

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
$$7$$ B.1.4

$p$-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 $$0$$.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 302330.bo consists of 2 curves linked by isogenies of degree 7.

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 3.1.172760.3 $$\Z/2\Z$$ Not in database
$$\Q(\zeta_{7})^+$$ $$\Z/7\Z$$ Not in database
6 $$x^{6} - 78 x^{4} + 1521 x^{2} + 172760$$ $$\Z/2\Z \times \Z/2\Z$$ 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.