Properties

 Label 303450.gp5 Conductor 303450 Discriminant 32555084678430720000000 j-invariant $$\frac{353108405631241}{86318776320}$$ CM no Rank 0 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -10638963, -10151783583]) # or

sage: E = EllipticCurve("303450gp3")

gp: E = ellinit([1, 0, 0, -10638963, -10151783583]) \\ or

gp: E = ellinit("303450gp3")

magma: E := EllipticCurve([1, 0, 0, -10638963, -10151783583]); // or

magma: E := EllipticCurve("303450gp3");

$$y^2 + x y = x^{3} - 10638963 x - 10151783583$$

Mordell-Weil group structure

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

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(3662, -1831\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(3662, -1831\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$303450$$ = $$2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$32555084678430720000000$$ = $$2^{24} \cdot 3 \cdot 5^{7} \cdot 7^{3} \cdot 17^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{353108405631241}{86318776320}$$ = $$2^{-24} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-3} \cdot 13^{3} \cdot 5437^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0850846864935$$ 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: $$288$$  = $$( 2^{3} \cdot 3 )\cdot1\cdot2\cdot3\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 303450.2.a.gp

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);

$$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + q^{12} - 2q^{13} + q^{14} + q^{16} + q^{18} + 8q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 31850496 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$6.12609742753$$

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$$(j)_{-}$$
$$2$$ $$24$$ $$I_{24}$$ Split multiplicative -1 1 24 24
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$2$$ $$I_1^{*}$$ Additive 1 2 7 1
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$17$$ $$2$$ $$I_0^{*}$$ 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 X13.

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

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
$$2$$ B
$$3$$ B

$p$-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 $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 303450.gp consists of 8 curves linked by isogenies of degrees dividing 12.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{17})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-255})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{105})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{1785})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\sqrt{-15}, \sqrt{17})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{105}, \sqrt{-119})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$\Q(\sqrt{-7}, \sqrt{-255})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{17}, \sqrt{105})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
6 6.2.906586678125.1 $$\Z/6\Z$$ 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.