Properties

Label 115608.a1
Conductor \(115608\)
Discriminant \(-343361146407936\)
j-invariant \( -\frac{46001912569921444}{335313619539} \)
CM no
Rank \(0\)
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -75240, -7968516]); // or
magma: E := EllipticCurve("115608a1");
sage: E = EllipticCurve([0, -1, 0, -75240, -7968516]) # or
sage: E = EllipticCurve("115608a1")
gp: E = ellinit([0, -1, 0, -75240, -7968516]) \\ or
gp: E = ellinit("115608a1")

\( y^2 = x^{3} - x^{2} - 75240 x - 7968516 \)

Mordell-Weil group structure

Trivial

Integral points

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 115608 \)  =  \(2^{3} \cdot 3 \cdot 4817\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(-343361146407936 \)  =  \(-1 \cdot 2^{10} \cdot 3 \cdot 4817^{3} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( -\frac{46001912569921444}{335313619539} \)  =  \(-1 \cdot 2^{2} \cdot 3^{-1} \cdot 4817^{-3} \cdot 225721^{3}\)
\( \text{End} (E) \)  =  \(\Z\)   (no Complex Multiplication)
\( \text{ST} (E) \)  =  $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
\( r \)  =  \(0\)
magma: Regulator(E);
sage: E.regulator()
\( \text{Reg} \)  =  \(1\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(0.144068095164\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 2 \)  = \( 2\cdot1\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
\( \#E_{\text{tor}} \)  = \(1\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Ш\(_{\text{an}} \)  =   \(1\) (exact)

Modular invariants

Modular form 115608.2.1.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

\( q - q^{3} + q^{5} + q^{9} - 5q^{11} + 4q^{13} - q^{15} - 7q^{17} + 8q^{19} + O(q^{20}) \)

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
714144 : curve is \( \Gamma_0(N) \)-optimal

Special L-value attached to the curve

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])

\( L(E,1) \) ≈ \( 0.288136190328 \)

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\((j)_{-}\)
\(2\) \(2\) \( III^{*} \) Additive -1 3 10 0
\(3\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(4817\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 \( p \) .

$p$-adic data

$p$-adic regulators

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

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 4817
Reduction type add nonsplit ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit
$\lambda$-invariant(s) - 0 2 0,0 0 0 0 0 0 0 0 0 0 2 0 0
$\mu$-invariant(s) - 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 no rational isogenies. Its isogeny class 115608.a consists of this curve only.