Properties

Label 348082.v1
Conductor \(348082\)
Discriminant \(305263317240668236423364608\)
j-invariant \( \frac{267635230617574617998257}{2062089938478825472} \)
CM no
Rank \(0\)
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -710224302, -7236834692117]); // or
magma: E := EllipticCurve("348082v1");
sage: E = EllipticCurve([1, 1, 1, -710224302, -7236834692117]) # or
sage: E = EllipticCurve("348082v1")
gp: E = ellinit([1, 1, 1, -710224302, -7236834692117]) \\ or
gp: E = ellinit("348082v1")

\( y^2 + x y + y = x^{3} + x^{2} - 710224302 x - 7236834692117 \)

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]
\( N \)  =  \( 348082 \)  =  \(2 \cdot 7 \cdot 23^{2} \cdot 47\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(305263317240668236423364608 \)  =  \(2^{20} \cdot 7^{7} \cdot 23^{7} \cdot 47^{3} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( \frac{267635230617574617998257}{2062089938478825472} \)  =  \(2^{-20} \cdot 7^{-7} \cdot 23^{-1} \cdot 47^{-3} \cdot 97^{3} \cdot 664369^{3}\)
\( \text{End} (E) \)  =  \(\Z\)   (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]
\( \Omega \)  ≈  \(0.0292585250701\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 120 \)  = \( ( 2^{2} \cdot 5 )\cdot1\cdot2\cdot3 \)
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 348082.2.1.v

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)

\( q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} - q^{7} + q^{8} - 2q^{9} + 2q^{10} + 6q^{11} - q^{12} - q^{13} - q^{14} - 2q^{15} + q^{16} - 7q^{17} - 2q^{18} - q^{19} + O(q^{20}) \)

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
223534080 : 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])

\( L(E,1) \) ≈ \( 3.51102300842 \)

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\((j)_{-}\)
\(2\) \(20\) \( I_{20} \) Split multiplicative -1 1 20 20
\(7\) \(1\) \( I_{7} \) Non-split multiplicative 1 1 7 7
\(23\) \(2\) \( I_1^{*} \) Additive -1 2 7 1
\(47\) \(3\) \( I_{3} \) 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 \(0\).

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has no rational isogenies. Its isogeny class 348082.v consists of this curve only.