Properties

Base field \(\Q(\sqrt{5}, \sqrt{17})\)
Label 4.4.7225.1-236.3-a2
Conductor \((118,\frac{7}{6} a^{3} - \frac{1}{2} a^{2} - \frac{77}{6} a + 9)\)
Conductor norm \( 236 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field \(\Q(\sqrt{5}, \sqrt{17})\)

Generator \(a\), with minimal polynomial \( x^{4} - 11 x^{2} + 9 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, -11, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 11*x^2 + 9)
gp (2.8): K = nfinit(a^4 - 11*a^2 + 9);

Weierstrass equation

\( y^2 + \left(-\frac{1}{6} a^{3} + \frac{1}{2} a^{2} + \frac{11}{6} a - 2\right) x y + \left(-\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{3}{2}\right) y = x^{3} + \left(\frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{11}{6} a + 3\right) x^{2} + \left(\frac{2}{3} a^{3} - a^{2} - \frac{22}{3} a + 8\right) x - a^{3} - 6 a^{2} - 6 a + 9 \)
magma: E := ChangeRing(EllipticCurve([-1/6*a^3 + 1/2*a^2 + 11/6*a - 2, 1/6*a^3 - 1/2*a^2 - 11/6*a + 3, -1/6*a^3 + 7/3*a + 3/2, 2/3*a^3 - a^2 - 22/3*a + 8, -a^3 - 6*a^2 - 6*a + 9]),K);
sage: E = EllipticCurve(K, [-1/6*a^3 + 1/2*a^2 + 11/6*a - 2, 1/6*a^3 - 1/2*a^2 - 11/6*a + 3, -1/6*a^3 + 7/3*a + 3/2, 2/3*a^3 - a^2 - 22/3*a + 8, -a^3 - 6*a^2 - 6*a + 9])
gp (2.8): E = ellinit([-1/6*a^3 + 1/2*a^2 + 11/6*a - 2, 1/6*a^3 - 1/2*a^2 - 11/6*a + 3, -1/6*a^3 + 7/3*a + 3/2, 2/3*a^3 - a^2 - 22/3*a + 8, -a^3 - 6*a^2 - 6*a + 9],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((118,\frac{7}{6} a^{3} - \frac{1}{2} a^{2} - \frac{77}{6} a + 9)\) = \( \left(2, -\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{1}{2}\right) \cdot \left(59, \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{47}{2}\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 236 \) = \( 4 \cdot 59 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((24234722,\frac{1}{3} a^{3} - \frac{8}{3} a + 5570257,-\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{39033897}{2},\frac{1}{6} a^{3} + \frac{1}{2} a^{2} - \frac{11}{6} a + 3374161)\) = \( \left(2, -\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{1}{2}\right) \cdot \left(59, \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{47}{2}\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 48469444 \) = \( 4 \cdot 59^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{35867214803}{48469444} a^{3} - \frac{15888032143}{48469444} a^{2} - \frac{455784216367}{48469444} a + \frac{114223634126}{12117361} \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

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

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]
Generator: $\left(-\frac{1}{24} a^{3} + \frac{3}{8} a^{2} + \frac{29}{24} a - 2 : -\frac{5}{16} a^{3} - \frac{1}{8} a^{2} + \frac{3}{2} a - \frac{45}{16} : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[3]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(59, \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{47}{2}\right) \) \(59\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(2, -\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{1}{2}\right) \) \(4\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 236.3-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.