# Properties

 Label 24563d1 Conductor $24563$ Discriminant $-2972123$ j-invariant $$-\frac{495616}{203}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -40, 142])

gp: E = ellinit([0, -1, 1, -40, 142])

magma: E := EllipticCurve([0, -1, 1, -40, 142]);

$$y^2+y=x^3-x^2-40x+142$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(4, 5\right)$$ $$\left(26, 126\right)$$ $$\hat{h}(P)$$ ≈ $0.20780321648266327591836258891$ $1.7099363549768523662489035470$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-7, 5\right)$$, $$\left(-7, -6\right)$$, $$\left(-2, 14\right)$$, $$\left(-2, -15\right)$$, $$\left(4, 5\right)$$, $$\left(4, -6\right)$$, $$\left(5, 6\right)$$, $$\left(5, -7\right)$$, $$\left(7, 12\right)$$, $$\left(7, -13\right)$$, $$\left(26, 126\right)$$, $$\left(26, -127\right)$$, $$\left(136, 1578\right)$$, $$\left(136, -1579\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$24563$$ = $$7 \cdot 11^{2} \cdot 29$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-2972123$$ = $$-1 \cdot 7 \cdot 11^{4} \cdot 29$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{495616}{203}$$ = $$-1 \cdot 2^{12} \cdot 7^{-1} \cdot 11^{2} \cdot 29^{-1}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$-0.049577609512475980748917412578\dots$$ Stable Faltings height: $$-0.84887603377859949543623193857\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.34672163679239327874917088809\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$2.3784289728759755167315766719\dots$$ 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: $$3$$  = $$1\cdot3\cdot1$$ 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 24563.2.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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 7872 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$2.4739583594100269650390041286730708774$$

## 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$$(j)_{-}$$
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$11$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$29$$ $$1$$ $$I_{1}$$ 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(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 ss ordinary ordinary nonsplit add ordinary ordinary ordinary ss nonsplit ordinary ordinary ordinary ordinary ordinary 2,9 2 6 2 - 2 2 2 2,2 2 2 2 2 2 2 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 no rational isogenies. Its isogeny class 24563d consists of this curve only.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.98252.2 $$\Z/2\Z$$ Not in database $6$ 6.0.1959651467312.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.