Properties

Label 124950de4
Conductor 124950
Discriminant 49744945778062500
j-invariant \( \frac{576615941610337}{27060804} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -2124176, 1191383498]); // or
 
magma: E := EllipticCurve("124950de4");
 
sage: E = EllipticCurve([1, 0, 1, -2124176, 1191383498]) # or
 
sage: E = EllipticCurve("124950de4")
 
gp: E = ellinit([1, 0, 1, -2124176, 1191383498]) \\ or
 
gp: E = ellinit("124950de4")
 

\( y^2 + x y + y = x^{3} - 2124176 x + 1191383498 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-1564, 27021\right) \)\( \left(816, 841\right) \)
\(\hat{h}(P)\) ≈  3.726178026840.456842079446

Torsion generators

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

\( \left(837, -419\right) \), \( \left(\frac{3383}{4}, -\frac{3387}{8}\right) \)

Integral points

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

\( \left(-1683, 841\right) \), \( \left(-1564, 27021\right) \), \( \left(-633, 48091\right) \), \( \left(-213, 40531\right) \), \( \left(102, 31186\right) \), \( \left(767, 3291\right) \), \( \left(816, 841\right) \), \( \left(837, -419\right) \), \( \left(852, 61\right) \), \( \left(858, 379\right) \), \( \left(867, 841\right) \), \( \left(918, 3442\right) \), \( \left(1037, 9681\right) \), \( \left(1152, 15961\right) \), \( \left(1887, 61531\right) \), \( \left(2652, 117886\right) \), \( \left(4386, 273946\right) \), \( \left(8517, 770941\right) \), \( \left(12087, 1313581\right) \), \( \left(51867, 11781841\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 124950 \)  =  \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 17\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(49744945778062500 \)  =  \(2^{2} \cdot 3^{4} \cdot 5^{6} \cdot 7^{6} \cdot 17^{4} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{576615941610337}{27060804} \)  =  \(2^{-2} \cdot 3^{-4} \cdot 17^{-4} \cdot 83233^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(2\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.971719181489\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.335858167885\)
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: \( 512 \)  = \( 2\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2} \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(4\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 124950.2.a.cp

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)
 

\( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - 4q^{11} + q^{12} - 2q^{13} + q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 3145728
\( \Gamma_0(N) \)-optimal: no
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])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 10.4435143678 \)

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\((j)_{-}\)
\(2\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(3\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(5\) \(4\) \( I_0^{*} \) Additive 1 2 6 0
\(7\) \(4\) \( I_0^{*} \) Additive -1 2 6 0
\(17\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4

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 X203.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 3 \end{array}\right)$ and has index 48.

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
\(2\) Cs

$p$-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 split add add ordinary ordinary split ordinary ss ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 9 3 - - 2 2 3 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

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 and 4.
Its isogeny class 124950de consists of 6 curves linked by isogenies of degrees dividing 8.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{35}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
4 \(\Q(\sqrt{2}, \sqrt{-35})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-2}, \sqrt{-35})\) \(\Z/2\Z \times \Z/4\Z\) 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.