# Properties

 Label 24301.a1 Conductor 24301 Discriminant -24301 j-invariant $$-\frac{9434056897}{24301}$$ CM no Rank 3 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -44, 109]) # or

sage: E = EllipticCurve("24301a1")

gp: E = ellinit([1, 0, 0, -44, 109]) \\ or

gp: E = ellinit("24301a1")

magma: E := EllipticCurve([1, 0, 0, -44, 109]); // or

magma: E := EllipticCurve("24301a1");

$$y^2 + x y = x^{3} - 44 x + 109$$

## Mordell-Weil group structure

$$\Z^3$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(3, -4\right)$$ $$\left(-5, 17\right)$$ $$\left(-4, 17\right)$$ $$\hat{h}(P)$$ ≈ 1.0100907588240366 2.2577453215579055 2.1784352519076147

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-5, 17\right)$$, $$\left(-5, -12\right)$$, $$\left(-4, 17\right)$$, $$\left(-4, -13\right)$$, $$\left(3, 1\right)$$, $$\left(3, -4\right)$$, $$\left(4, -1\right)$$, $$\left(4, -3\right)$$, $$\left(5, 2\right)$$, $$\left(5, -7\right)$$, $$\left(7, 9\right)$$, $$\left(7, -16\right)$$, $$\left(9, 17\right)$$, $$\left(9, -26\right)$$, $$\left(21, 82\right)$$, $$\left(21, -103\right)$$, $$\left(28, 131\right)$$, $$\left(28, -159\right)$$, $$\left(60, 433\right)$$, $$\left(60, -493\right)$$, $$\left(879, 25624\right)$$, $$\left(879, -26503\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$24301$$ = $$19 \cdot 1279$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-24301$$ = $$-1 \cdot 19 \cdot 1279$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{9434056897}{24301}$$ = $$-1 \cdot 19^{-1} \cdot 1279^{-1} \cdot 2113^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$3$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.08856460243$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$3.79628009247$$ 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: $$1$$  = $$1\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 24301.2.a.a

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3584 $$\Gamma_0(N)$$-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);

$$L^{(3)}(E,1)/3!$$ ≈ $$4.13249612956$$

## Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$19$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$1279$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ .

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ 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 1279 ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary nonsplit 5 3 7 3 3 3 3 3 3 3 3 3,3 3 3 3 ? 0 0 0 0 0 0 0 0 0 0 0 0,0 0 0 0 ?

An entry ? indicates that the invariants have not yet been computed.

## Isogenies

This curve has no rational isogenies. Its isogeny class 24301.a consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.97204.1 $$\Z/2\Z$$ Not in database
6 $$x^{6} + 6 x^{4} + 9 x^{2} + 97204$$ $$\Z/2\Z \times \Z/2\Z$$ 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.