# Properties

 Label 24.a4 Conductor 24 Discriminant 2304 j-invariant $$\frac{35152}{9}$$ CM no Rank 0 Torsion Structure $$\Z/{2}\Z \times \Z/{4}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -4, 4]); // or
magma: E := EllipticCurve("24a1");
sage: E = EllipticCurve([0, -1, 0, -4, 4]) # or
sage: E = EllipticCurve("24a1")
gp: E = ellinit([0, -1, 0, -4, 4]) \\ or
gp: E = ellinit("24a1")

$$y^2 = x^{3} - x^{2} - 4 x + 4$$

## Mordell-Weil group structure

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

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

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

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

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

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$24$$ = $$2^{3} \cdot 3$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$2304$$ = $$2^{8} \cdot 3^{2}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{35152}{9}$$ = $$2^{4} \cdot 3^{-2} \cdot 13^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$0$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$4.313031295$$ 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: $$8$$  = $$2^{2}\cdot2$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$8$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form24.2.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)

$$q - q^{3} - 2q^{5} + q^{9} + 4q^{11} - 2q^{13} + 2q^{15} + 2q^{17} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 1 $$\Gamma_0(N)$$-optimal: yes 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])

$$L(E,1)$$ ≈ $$0.539128911875$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_1^{*}$$ Additive -1 3 8 0
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## 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 X190c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 0 \\ 4 & 5 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 5 \end{array}\right)$ and has index 96.

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
$$2$$ Cs

## $p$-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 $$0$$.

## 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 up to 7 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:

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