Properties

 Label 24336i6 Conductor $24336$ Discriminant $6.486\times 10^{13}$ j-invariant $$\frac{3065617154}{9}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -584571, -172029494])

gp: E = ellinit([0, 0, 0, -584571, -172029494])

magma: E := EllipticCurve([0, 0, 0, -584571, -172029494]);

$$y^2=x^3-584571x-172029494$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-441, 14\right)$$ $$\left(9359, 902286\right)$$ $$\hat{h}(P)$$ ≈ $2.9700369171280745358731764119$ $7.3984216539745364771279580185$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-442, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-442, 0\right)$$, $$(-441,\pm 14)$$, $$(1079,\pm 21294)$$, $$(9359,\pm 902286)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$24336$$ = $$2^{4} \cdot 3^{2} \cdot 13^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$64857485002752$$ = $$2^{11} \cdot 3^{8} \cdot 13^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3065617154}{9}$$ = $$2 \cdot 3^{-2} \cdot 1153^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.8795757175449112504825877839\dots$$ Stable Faltings height: $$-0.58759002103319516354090800001\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$20.378633691142493309633305836\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.17265943464294997020350483192\dots$$ 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: $$8$$  = $$2\cdot2\cdot2$$ 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: $$1$$ (rounded)

Modular invariants

Modular form 24336.2.a.o

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);

$$q - 2q^{5} - 4q^{11} - 2q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 147456 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$7.0371267438168712653965581926473643492$$

Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_3^{*}$$ Additive 1 4 11 0
$$3$$ $$2$$ $$I_2^{*}$$ Additive -1 2 8 2
$$13$$ $$2$$ $$I_0^{*}$$ 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 X234.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 3 \end{array}\right)$ and has index 48.

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

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 add add ordinary ss ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss - - 2 2,2 2 - 2 2 2 2 4 2 2 2 2,2 - - 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, 4 and 8.
Its isogeny class 24336i consists of 4 curves linked by isogenies of degrees dividing 8.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-78})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-39})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-39})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{-26})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{-13})$$ $$\Z/8\Z$$ Not in database $8$ 8.4.38813099360256.25 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.38813099360256.44 $$\Z/8\Z$$ Not in database $8$ 8.0.151613669376.5 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.5458092097536.9 $$\Z/16\Z$$ Not in database $8$ 8.2.20723693432832.1 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.