Properties

 Label 3360.p3 Conductor $3360$ Discriminant $226842638400$ j-invariant $$\frac{13507798771700416}{3544416225}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -19846, 1069280])

gp: E = ellinit([0, 1, 0, -19846, 1069280])

magma: E := EllipticCurve([0, 1, 0, -19846, 1069280]);

$$y^2=x^3+x^2-19846x+1069280$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(26, 756\right)$$ $$\hat{h}(P)$$ ≈ $0.39414992061779604880052647032$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(80, 0\right)$$, $$\left(82, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-163, 0\right)$$, $$(-16,\pm 1176)$$, $$(26,\pm 756)$$, $$(77,\pm 60)$$, $$\left(80, 0\right)$$, $$\left(82, 0\right)$$, $$(89,\pm 126)$$, $$(152,\pm 1260)$$, $$(242,\pm 3240)$$, $$(2105,\pm 96390)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$3360$$ = $$2^{5} \cdot 3 \cdot 5 \cdot 7$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$226842638400$$ = $$2^{6} \cdot 3^{10} \cdot 5^{2} \cdot 7^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{13507798771700416}{3544416225}$$ = $$2^{6} \cdot 3^{-10} \cdot 5^{-2} \cdot 7^{-4} \cdot 59539^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.1636834470806870687954653800\dots$$ Stable Faltings height: $$0.81710985680071441408684931927\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.39414992061779604880052647032\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.97008785814110665130710830845\dots$$ 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: $$160$$  = $$2\cdot( 2 \cdot 5 )\cdot2\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

Modular invariants

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);

$$q + q^{3} - q^{5} + q^{7} + q^{9} - 4q^{11} - 6q^{13} - q^{15} + 6q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 7680 $$\Gamma_0(N)$$-optimal: yes 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);

$$L'(E,1)$$ ≈ $$3.8236005227860498107310595970771997616$$

Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

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$$(j)_{-}$$
$$2$$ $$2$$ $$III$$ Additive 1 5 6 0
$$3$$ $$10$$ $$I_{10}$$ Split multiplicative -1 1 10 10
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4

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 X8c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ and has index 12.

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
$$2$$ Cs

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ 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 add split nonsplit split ordinary ordinary ordinary ss ordinary ordinary ss ordinary ordinary ordinary ordinary - 6 1 2 1 1 1 1,1 1 1 1,1 1 1 1 1 - 0 0 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 3360.p consists of 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\sqrt{-3}, \sqrt{-5})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{5})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{3})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ 16.0.11007531417600000000.1 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.