Properties

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

Related objects

Downloads

Learn more about

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]);
 

\(y^2+y=x^3-x^2-40x+142\)  Toggle raw display

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

sage: E.gens()
 
magma: Generators(E);
 

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

Integral points

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

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

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);
 

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

For more coefficients, see the Downloads section to the right.

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ordinary ordinary nonsplit add ordinary ordinary ordinary ss nonsplit ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2,9 2 6 2 - 2 2 2 2,2 2 2 2 2 2 2
$\mu$-invariant(s) 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 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.