Base field 5.5.65657.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1)
gp (2.8): K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 - a - 1, -a^2 + a + 2, -a^4 + 2*a^3 + 4*a^2 - 6*a - 3, 4*a^4 - 5*a^3 - 19*a^2 + 13*a + 17, 8*a^4 - 10*a^3 - 37*a^2 + 25*a + 32]),K);
sage: E = EllipticCurve(K, [a^2 - a - 1, -a^2 + a + 2, -a^4 + 2*a^3 + 4*a^2 - 6*a - 3, 4*a^4 - 5*a^3 - 19*a^2 + 13*a + 17, 8*a^4 - 10*a^3 - 37*a^2 + 25*a + 32])
gp (2.8): E = ellinit([a^2 - a - 1, -a^2 + a + 2, -a^4 + 2*a^3 + 4*a^2 - 6*a - 3, 4*a^4 - 5*a^3 - 19*a^2 + 13*a + 17, 8*a^4 - 10*a^3 - 37*a^2 + 25*a + 32],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((45,-a^{3} + 4 a)\) | = | \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{2} \cdot \left(-a^{2} + a + 2\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 45 \) | = | \( 3^{2} \cdot 5 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((2278125,a + 1754318,a^{4} - a^{3} - 4 a^{2} + 2 a + 2177699,-a^{4} + a^{3} + 5 a^{2} - 3 a + 1947859,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 2197336)\) | = | \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{6} \cdot \left(-a^{2} + a + 2\right)^{5} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 2278125 \) | = | \( 3^{6} \cdot 5^{5} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{81829946}{3125} a^{4} - \frac{143808649}{3125} a^{3} - \frac{300612723}{3125} a^{2} + \frac{393132181}{3125} a + \frac{109010922}{3125} \) | ||
| 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: | \(\Z/3\Z\) |
|---|---|
| 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: | $\left(0 : 2 a^{4} - 3 a^{3} - 9 a^{2} + 8 a + 8 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) | \(3\) | \(1\) | \(I_{0}^*\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \( \left(-a^{2} + a + 2\right) \) | \(5\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
| \(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
45.1-b
consists of curves linked by isogenies of
degrees dividing 15.