Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-23.1-a2
Conductor \((23,-a^{4} + 3 a^{2} + 1)\)
Conductor norm \( 23 \)
CM no
base-change no
Q-curve no
Torsion order \( 10 \)
Rank not available

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Base field \(\Q(\zeta_{11})^+\)

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1)
gp (2.8): K = nfinit(a^5 - a^4 - 4*a^3 + 3*a^2 + 3*a - 1);

Weierstrass equation

\( y^2 + \left(a^{3} + a^{2} - 2 a - 2\right) x y + \left(a^{3} - 3 a + 1\right) y = x^{3} + \left(-a^{4} - a^{3} + 5 a^{2} + 4 a - 4\right) x^{2} + \left(-5 a^{4} + 7 a^{3} + 11 a^{2} - 3 a - 4\right) x - a^{4} - 13 a^{3} + 27 a^{2} + 14 a - 8 \)
magma: E := ChangeRing(EllipticCurve([a^3 + a^2 - 2*a - 2, -a^4 - a^3 + 5*a^2 + 4*a - 4, a^3 - 3*a + 1, -5*a^4 + 7*a^3 + 11*a^2 - 3*a - 4, -a^4 - 13*a^3 + 27*a^2 + 14*a - 8]),K);
sage: E = EllipticCurve(K, [a^3 + a^2 - 2*a - 2, -a^4 - a^3 + 5*a^2 + 4*a - 4, a^3 - 3*a + 1, -5*a^4 + 7*a^3 + 11*a^2 - 3*a - 4, -a^4 - 13*a^3 + 27*a^2 + 14*a - 8])
gp (2.8): E = ellinit([a^3 + a^2 - 2*a - 2, -a^4 - a^3 + 5*a^2 + 4*a - 4, a^3 - 3*a + 1, -5*a^4 + 7*a^3 + 11*a^2 - 3*a - 4, -a^4 - 13*a^3 + 27*a^2 + 14*a - 8],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((23,-a^{4} + 3 a^{2} + 1)\) = \( \left(a^{4} - 3 a^{2} - 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 23 \) = \( 23 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((41426511213649,a + 28012496092748,a^{2} + 16402527694628,a^{3} - 3 a + 6450908953021,a^{4} - 4 a^{2} + 6901437066764)\) = \( \left(a^{4} - 3 a^{2} - 1\right)^{10} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 41426511213649 \) = \( 23^{10} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{74868976618071917973}{41426511213649} a^{4} - \frac{54092178455960064989}{41426511213649} a^{3} - \frac{312961789930170680193}{41426511213649} a^{2} + \frac{134824139137681173208}{41426511213649} a + \frac{261494050662338262657}{41426511213649} \)
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: \(\Z/10\Z\)
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: $\left(-9 a^{4} + 8 a^{3} + 26 a^{2} - 18 a + 2 : -39 a^{4} + 34 a^{3} + 119 a^{2} - 82 a + 12 : 1\right)$
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\((j)_{-}\)
\( \left(a^{4} - 3 a^{2} - 1\right) \) \(23\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 23.1-a consists of curves linked by isogenies of degrees dividing 10.

Base change

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