Properties

Label 8036.e1
Conductor 8036
Discriminant 7597454297104
j-invariant \( \frac{373698304}{1681} \)
CM no
Rank 0
Torsion Structure \(\mathrm{Trivial}\)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -24810, -1506583]); // or
 
magma: E := EllipticCurve("8036e1");
 
sage: E = EllipticCurve([0, 1, 0, -24810, -1506583]) # or
 
sage: E = EllipticCurve("8036e1")
 
gp: E = ellinit([0, 1, 0, -24810, -1506583]) \\ or
 
gp: E = ellinit("8036e1")
 

\( y^2 = x^{3} + x^{2} - 24810 x - 1506583 \)

Mordell-Weil group structure

Trivial

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 
None

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 8036 \)  =  \(2^{2} \cdot 7^{2} \cdot 41\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(7597454297104 \)  =  \(2^{4} \cdot 7^{10} \cdot 41^{2} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{373698304}{1681} \)  =  \(2^{8} \cdot 7^{2} \cdot 31^{3} \cdot 41^{-2}\)
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: \(0.380503761866\)
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: \( 6 \)  = \( 3\cdot1\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 8036.2.a.e

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)
 

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

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 17136
\( \Gamma_0(N) \)-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])
 

\( L(E,1) \) ≈ \( 2.2830225712 \)

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\((j)_{-}\)
\(2\) \(3\) \( IV \) Additive -1 2 4 0
\(7\) \(1\) \( II^{*} \) Additive -1 2 10 0
\(41\) \(2\) \( I_{2} \) 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 X2.

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

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
\(2\) Cn

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split ordinary ordinary
$\lambda$-invariant(s) - 4 4 - 0 0 0 0 0 0 0 0 1 0 0
$\mu$-invariant(s) - 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 8036.e 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 \(\Q(\zeta_{7})^+\) \(\Z/2\Z \times \Z/2\Z\) 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.