Properties

 Label 3315b4 Conductor 3315 Discriminant -910381875 j-invariant $$\frac{688699320191}{910381875}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, 184, -1012]); // or

magma: E := EllipticCurve("3315b4");

sage: E = EllipticCurve([1, 1, 1, 184, -1012]) # or

sage: E = EllipticCurve("3315b4")

gp: E = ellinit([1, 1, 1, 184, -1012]) \\ or

gp: E = ellinit("3315b4")

$$y^2 + x y + y = x^{3} + x^{2} + 184 x - 1012$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(8, 28\right)$$ $$\left(11, 44\right)$$ $$\hat{h}(P)$$ ≈ 1.00673497113 1.44468860936

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(\frac{19}{4}, -\frac{23}{8}\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(8, 28\right)$$, $$\left(8, -37\right)$$, $$\left(11, 44\right)$$, $$\left(11, -56\right)$$, $$\left(34, 197\right)$$, $$\left(34, -232\right)$$, $$\left(47, 314\right)$$, $$\left(47, -362\right)$$, $$\left(86, 769\right)$$, $$\left(86, -856\right)$$, $$\left(736, 19619\right)$$, $$\left(736, -20356\right)$$, $$\left(1724, 70748\right)$$, $$\left(1724, -72473\right)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$3315$$ = $$3 \cdot 5 \cdot 13 \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-910381875$$ = $$-1 \cdot 3 \cdot 5^{4} \cdot 13^{4} \cdot 17$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{688699320191}{910381875}$$ = $$3^{-1} \cdot 5^{-4} \cdot 13^{-4} \cdot 17^{-1} \cdot 8831^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.35634345933$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.839494440253$$ 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: $$8$$  = $$1\cdot2\cdot2^{2}\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form3315.2.a.a

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)

$$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - 4q^{7} + 3q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} + q^{13} + 4q^{14} + q^{15} - q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1792 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$2.27728558637$$

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$$(j)_{-}$$
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

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.

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
$$2$$ B

$p$-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 ordinary nonsplit nonsplit ordinary ordinary split nonsplit ordinary ss ordinary ss ordinary ordinary ordinary ordinary 3 2 2 2 2 3 2 2 2,2 2 2,2 2 2 2 2 2 0 0 0 0 0 0 0 0,0 0 0,0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 3315b consists of 4 curves linked by isogenies of degrees dividing 4.

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 $$\Q(\sqrt{3})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-51})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{-17})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\sqrt{3}, \sqrt{-17})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.