Properties

Label 191466.t1
Conductor \(191466\)
Discriminant \(12659686629866496\)
j-invariant \( \frac{30154864531691593}{17365825281024} \)
CM no
Rank \(0\)
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -58361, 393113]); // or
magma: E := EllipticCurve("191466j1");
sage: E = EllipticCurve([1, -1, 1, -58361, 393113]) # or
sage: E = EllipticCurve("191466j1")
gp: E = ellinit([1, -1, 1, -58361, 393113]) \\ or
gp: E = ellinit("191466j1")

\( y^2 + x y + y = x^{3} - x^{2} - 58361 x + 393113 \)

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]
\( N \)  =  \( 191466 \)  =  \(2 \cdot 3^{2} \cdot 11 \cdot 967\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(12659686629866496 \)  =  \(2^{10} \cdot 3^{19} \cdot 11 \cdot 967 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( \frac{30154864531691593}{17365825281024} \)  =  \(2^{-10} \cdot 3^{-13} \cdot 11^{-1} \cdot 29^{3} \cdot 967^{-1} \cdot 10733^{3}\)
\( \text{End} (E) \)  =  \(\Z\)   (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]
\( \Omega \)  ≈  \(0.340621805271\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 20 \)  = \( ( 2 \cdot 5 )\cdot2\cdot1\cdot1 \)
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 191466.2.1.t

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)

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
1497600 : 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])

\( L(E,1) \) ≈ \( 6.81243610541 \)

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\((j)_{-}\)
\(2\) \(10\) \( I_{10} \) Split multiplicative -1 1 10 10
\(3\) \(2\) \( I_13^{*} \) Additive -1 2 19 13
\(11\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(967\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

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 is surjective for all primes \( p \).

p-adic data

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 \(0\).