Properties

Label 348726n1
Conductor $348726$
Discriminant $-1.553\times 10^{22}$
j-invariant \( -\frac{319620691295711664553}{914283853056} \)
CM no
Rank $1$
Torsion structure \(\Z/{3}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -366139565, 2696585928680])
 
gp: E = ellinit([1, 0, 1, -366139565, 2696585928680])
 
magma: E := EllipticCurve([1, 0, 1, -366139565, 2696585928680]);
 

\(y^2+xy+y=x^3-366139565x+2696585928680\)

Mordell-Weil group structure

\(\Z\times \Z/{3}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(11427, 63838\right) \)
\(\hat{h}(P)\) ≈  $1.2371797458664921382986289258$

Torsion generators

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

\( \left(10860, 28684\right) \)

Integral points

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

\( \left(-18381, 1802638\right) \), \( \left(-18381, -1784258\right) \), \( \left(10860, 28684\right) \), \( \left(10860, -39545\right) \), \( \left(11427, 63838\right) \), \( \left(11427, -75266\right) \), \( \left(69342, 17592778\right) \), \( \left(69342, -17662121\right) \)

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: \(-15527797455744956503296 \)  =  \(-1 \cdot 2^{8} \cdot 3^{9} \cdot 7^{3} \cdot 19^{8} \cdot 23^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{319620691295711664553}{914283853056} \)  =  \(-1 \cdot 2^{-8} \cdot 3^{-9} \cdot 7^{-3} \cdot 19^{4} \cdot 23^{-2} \cdot 134857^{3}\)
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);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.2371797458664921382986289258\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.10805932310682600056016125669\)
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: \( 324 \)  = \( 2\cdot3^{2}\cdot3\cdot3\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(3\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 348726.2.a.n

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 106375680
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 4.8127970123930935106757455007038279308 \)

Local data

This elliptic curve is not semistable. There are 5 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\((j)_{-}\)
\(2\) \(2\) \(I_{8}\) Non-split multiplicative 1 1 8 8
\(3\) \(9\) \(I_{9}\) Split multiplicative -1 1 9 9
\(7\) \(3\) \(I_{3}\) Split multiplicative -1 1 3 3
\(19\) \(3\) \(IV^{*}\) Additive 1 2 8 0
\(23\) \(2\) \(I_{2}\) Non-split multiplicative 1 1 2 2

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 \( p \) except those listed.

prime Image of Galois representation
\(3\) B.1.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 \(d=\) 3.
Its isogeny class 348726n consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.30324.1 \(\Z/6\Z\) Not in database
$6$ 6.0.77241777984.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.15754676671152.1 \(\Z/3\Z \times \Z/3\Z\) Not in database
$9$ 9.3.487789519555497770542512.1 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$18$ 18.0.117775969925012931961921962728136757672229732352.1 \(\Z/3\Z \times \Z/6\Z\) 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.