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This is a model for the modular curve $X_0(24)$.

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -4, 4])

gp: E = ellinit([0, -1, 0, -4, 4])

magma: E := EllipticCurve([0, -1, 0, -4, 4]);

$$y^2=x^3-x^2-4x+4$$ Mordell-Weil group structure

$\Z/{2}\Z \times \Z/{4}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(1, 0\right)$$, $$\left(4, 6\right)$$ Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2, 0\right)$$, $$(0,\pm 2)$$, $$\left(1, 0\right)$$, $$\left(2, 0\right)$$, $$(4,\pm 6)$$ Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$24$$ = $2^{3} \cdot 3$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $2304$ = $2^{8} \cdot 3^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{35152}{9}$$ = $2^{4} \cdot 3^{-2} \cdot 13^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.64535228607985727265901067683\dots$ Stable Faltings height: $-1.1074504064531541456038320911\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $4.3130312949992864708773499976\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $8$  = $2^{2}\cdot2$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $8$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar/factorial(ar)  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $0.53912891187491080885966874970005048289$

Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{3} - 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1 $\Gamma_0(N)$-optimal: yes Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{1}^{*}$ Additive -1 3 8 0
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.96.0.42

$p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ Reduction type 2 3 add nonsplit - 0 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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 and 4.
Its isogeny class 24.a consists of 6 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\zeta_{12})$$ $$\Z/4\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{3})$$ $$\Z/2\Z \times \Z/8\Z$$ 4.4.2304.1-72.1-b4 $4$ $$\Q(\zeta_{8})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ $$\Q(\zeta_{24})$$ $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.0.2985984.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.2.181398528.1 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ 16.0.36520347436056576.1 $$\Z/8\Z \times \Z/8\Z$$ Not in database $16$ 16.8.2393397489569403764736.2 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ 16.0.364791569817010176.2 $$\Z/2\Z \times \Z/16\Z$$ Not in database

We only show fields where the torsion growth is primitive.

This curve is also a quotient of the genus-3 hyperelliptic curve $$y^2 = x^8 + 14x^4 + 1$$ which has geometric automorphism group $S_4 \times \Z/2\Z$.