Properties

Label 50430f5
Conductor $50430$
Discriminant $1.603\times 10^{14}$
j-invariant \( \frac{2656166199049}{33750} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -485003, 129803103]) # or
 
sage: E = EllipticCurve("50430f5")
 
gp: E = ellinit([1, 1, 0, -485003, 129803103]) \\ or
 
gp: E = ellinit("50430f5")
 
magma: E := EllipticCurve([1, 1, 0, -485003, 129803103]); // or
 
magma: E := EllipticCurve("50430f5");
 

\( y^2 + x y = x^{3} + x^{2} - 485003 x + 129803103 \)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{68371}{169}, -\frac{239049}{2197}\right) \)
\(\hat{h}(P)\) ≈  $7.605030123692327$

Torsion generators

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

\( \left(\frac{1611}{4}, -\frac{1611}{8}\right) \)

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 50430 \)  =  \(2 \cdot 3 \cdot 5 \cdot 41^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(160316018133750 \)  =  \(2 \cdot 3^{3} \cdot 5^{4} \cdot 41^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2656166199049}{33750} \)  =  \(2^{-1} \cdot 3^{-3} \cdot 5^{-4} \cdot 11^{3} \cdot 1259^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(7.60503012369\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.523486369302\)
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: \( 4 \)  = \( 1\cdot1\cdot2\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(2\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 50430.2.a.f

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 552960
\( \Gamma_0(N) \)-optimal: no
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);
 

\( L'(E,1) \) ≈ \( 3.98112960788 \)

Local data

This elliptic curve is not 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\((j)_{-}\)
\(2\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(3\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3
\(5\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(41\) \(2\) \( I_0^{*} \) Additive 1 2 6 0

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 X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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 \( p \) except those listed.

prime Image of Galois representation
\(2\) B
\(3\) B

$p$-adic data

$p$-adic regulators

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

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit nonsplit nonsplit ordinary ss ordinary ordinary ordinary ss ordinary ordinary ordinary add ordinary ss
$\lambda$-invariant(s) 4 1 5 1 1,1 3 1 1 1,1 1 1 1 - 1 1,1
$\mu$-invariant(s) 0 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 non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 50430f consists of 8 curves linked by isogenies of degrees dividing 12.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{6}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{41}) \) \(\Z/6\Z\) Not in database
$2$ \(\Q(\sqrt{82}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{123}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{41})\) \(\Z/12\Z\) Not in database
$4$ \(\Q(\sqrt{3}, \sqrt{41})\) \(\Z/12\Z\) Not in database
$4$ \(\Q(\sqrt{6}, \sqrt{41})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{6}, \sqrt{82})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$6$ 6.0.18608670000.8 \(\Z/6\Z\) 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.