Properties

Label 348726r2
Conductor $348726$
Discriminant $1.239\times 10^{18}$
j-invariant \( \frac{169967019783457}{26337394944} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -416602, -88598260]) # or
 
sage: E = EllipticCurve("348726r2")
 
gp: E = ellinit([1, 0, 1, -416602, -88598260]) \\ or
 
gp: E = ellinit("348726r2")
 
magma: E := EllipticCurve([1, 0, 1, -416602, -88598260]); // or
 
magma: E := EllipticCurve("348726r2");
 

\( y^2 + x y + y = x^{3} - 416602 x - 88598260 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(\frac{29701}{9}, -\frac{5058133}{27}\right) \)\( \left(2044, -88204\right) \)
\(\hat{h}(P)\) ≈  $4.613021045547634$$0.9901975745954401$

Torsion generators

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

\( \left(733, -367\right) \), \( \left(-483, 241\right) \)

Integral points

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

\( \left(-483, 241\right) \), \( \left(-467, 2273\right) \), \( \left(-467, -1807\right) \), \( \left(-371, 4049\right) \), \( \left(-371, -3679\right) \), \( \left(-350, 3965\right) \), \( \left(-350, -3616\right) \), \( \left(733, -367\right) \), \( \left(742, 2936\right) \), \( \left(742, -3679\right) \), \( \left(1645, 59825\right) \), \( \left(1645, -61471\right) \), \( \left(2044, 86159\right) \), \( \left(2044, -88204\right) \), \( \left(4620, 308500\right) \), \( \left(4620, -313121\right) \), \( \left(17206, 2246744\right) \), \( \left(17206, -2263951\right) \), \( \left(133581, 48754849\right) \), \( \left(133581, -48888431\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 348726 \)  =  \(2 \cdot 3 \cdot 7 \cdot 19^{2} \cdot 23\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(1239065948385425664 \)  =  \(2^{8} \cdot 3^{4} \cdot 7^{4} \cdot 19^{6} \cdot 23^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{169967019783457}{26337394944} \)  =  \(2^{-8} \cdot 3^{-4} \cdot 7^{-4} \cdot 13^{3} \cdot 23^{-2} \cdot 4261^{3}\)
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: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(3.97211455111\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.189877835101\)
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: \( 256 \)  = \( 2\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(4\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 348726.2.a.r

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);
 

\( q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + 2q^{10} - 4q^{11} + q^{12} + 2q^{13} - q^{14} - 2q^{15} + q^{16} - 6q^{17} - q^{18} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 7077888
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 12.0674641878 \)

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\((j)_{-}\)
\(2\) \(2\) \( I_{8} \) Non-split multiplicative 1 1 8 8
\(3\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(7\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(19\) \(4\) \( I_0^{*} \) Additive -1 2 6 0
\(23\) \(2\) \( I_{2} \) Non-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 X98.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 6 \\ 4 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 7 \end{array}\right)$ and has index 24.

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
\(2\) 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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 348726r 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
$2$ \(\Q(\sqrt{-19}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-7}, \sqrt{-19})\) \(\Z/2\Z \times \Z/8\Z\) Not in database
$4$ \(\Q(\sqrt{19}, \sqrt{-23})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{19}, \sqrt{23})\) \(\Z/2\Z \times \Z/4\Z\) 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.