Properties

Label 69366w2
Conductor 69366
Discriminant -373958095272087819200664
j-invariant \( -\frac{925492188434597796818942768449}{373958095272087819200664} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -203025076, -1113860484568]) # or
 
sage: E = EllipticCurve("69366w2")
 
gp: E = ellinit([1, 0, 0, -203025076, -1113860484568]) \\ or
 
gp: E = ellinit("69366w2")
 
magma: E := EllipticCurve([1, 0, 0, -203025076, -1113860484568]); // or
 
magma: E := EllipticCurve("69366w2");
 

\( y^2 + x y = x^{3} - 203025076 x - 1113860484568 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{24880589116471}{298944100}, -\frac{122366764350732057239}{5168743489000}\right) \)
\(\hat{h}(P)\) ≈  30.93937299196835

Integral points

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

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 69366 \)  =  \(2 \cdot 3 \cdot 11 \cdot 1051\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-373958095272087819200664 \)  =  \(-1 \cdot 2^{3} \cdot 3 \cdot 11 \cdot 1051^{7} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{925492188434597796818942768449}{373958095272087819200664} \)  =  \(-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 11^{-1} \cdot 43^{3} \cdot 97^{3} \cdot 127^{3} \cdot 1051^{-7} \cdot 18397^{3}\)
Endomorphism ring: \(\Z\)   (no 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: \(30.939372992\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.0199973115272\)
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: \( 21 \)  = \( 3\cdot1\cdot1\cdot7 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 69366.2.a.u

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);
 

\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + q^{14} - q^{15} + q^{16} + 4q^{17} + q^{18} + 6q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 13253520
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 12.9927898837 \)

Local data

This elliptic curve is 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\((j)_{-}\)
\(2\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(3\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(11\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(1051\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7

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
\(7\) B.1.3

$p$-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 1051
Reduction type split split ordinary ordinary split ss ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary split
$\lambda$-invariant(s) 3 4 1 3 4 1,1 5 1 1 1 1 1 1,1 1 1 ?
$\mu$-invariant(s) 0 0 0 1 0 0,0 0 0 0 0 0 0 0,0 0 0 ?

An entry ? indicates that the invariants have not yet been computed.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 69366w consists of 2 curves linked by isogenies of degree 7.

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 3.1.277464.4 \(\Z/2\Z\) Not in database
6 \(\Q(\zeta_{7})\) \(\Z/7\Z\) Not in database
\( x^{6} - 194 x^{4} + 9409 x^{2} + 1109856 \) \(\Z/2\Z \times \Z/2\Z\) Not in database
7 7.1.68069081958026688.1 \(\Z/7\Z\) 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.