Properties

Label 38.b2
Conductor \(38\)
Discriminant \(-608\)
j-invariant \( -\frac{1}{608} \)
CM no
Rank \(0\)
Torsion Structure \(\Z/{5}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, 0, 1]); // or
magma: E := EllipticCurve("38b1");
sage: E = EllipticCurve([1, 1, 1, 0, 1]) # or
sage: E = EllipticCurve("38b1")
gp: E = ellinit([1, 1, 1, 0, 1]) \\ or
gp: E = ellinit("38b1")

\( y^2 + x y + y = x^{3} + x^{2} + 1 \)

Mordell-Weil group structure

\(\Z/{5}\Z\)

Torsion generators

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

\( \left(1, 1\right) \)

Integral points

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

\( \left(-1, 1\right) \), \( \left(1, 1\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 38 \)  =  \(2 \cdot 19\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(-608 \)  =  \(-1 \cdot 2^{5} \cdot 19 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( -\frac{1}{608} \)  =  \(-1 \cdot 2^{-5} \cdot 19^{-1}\)
\( \text{End} (E) \)  =  \(\Z\)   (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]
\( \Omega \)  ≈  \(4.09622816528\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 5 \)  = \( 5\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
\( \#E_{\text{tor}} \)  = \(5\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Ш\(_{\text{an}} \)  =   \(1\) (exact)

Modular invariants

Modular form 38.2.1.b

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)

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
2 : 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])

\( L(E,1) \) ≈ \( 0.819245633055 \)

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\((j)_{-}\)
\(2\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(19\) \(1\) \( I_{1} \) 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 \( p \) except those listed.

prime Image of Galois representation
\(5\) B.1.1

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 \(0\).

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 38.b consists of 2 curves linked by isogenies of degrees dividing 5.