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 \(1\).
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1)
gp: K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve(K, [a^4 - a^3 - 4*a^2 + 2*a + 3, -3*a^4 + 4*a^3 + 13*a^2 - 10*a - 10, a^2 - a - 2, 4*a^4 - 3*a^3 - 20*a^2 + 5*a + 15, -11*a^4 + 49*a^2 + 27*a - 4])
gp: E = ellinit([a^4 - a^3 - 4*a^2 + 2*a + 3, -3*a^4 + 4*a^3 + 13*a^2 - 10*a - 10, a^2 - a - 2, 4*a^4 - 3*a^3 - 20*a^2 + 5*a + 15, -11*a^4 + 49*a^2 + 27*a - 4],K)
magma: E := ChangeRing(EllipticCurve([a^4 - a^3 - 4*a^2 + 2*a + 3, -3*a^4 + 4*a^3 + 13*a^2 - 10*a - 10, a^2 - a - 2, 4*a^4 - 3*a^3 - 20*a^2 + 5*a + 15, -11*a^4 + 49*a^2 + 27*a - 4]),K);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((25,-a^{3} + a^{2} + 3 a - 2)\) | = | \( \left(-a^{2} + a + 2\right)^{2} \) |
| sage: E.conductor()
magma: Conductor(E);
| ||||
| \(N(\mathfrak{N}) \) | = | \( 25 \) | = | \( 5^{2} \) |
| sage: E.conductor().norm()
magma: Norm(Conductor(E));
| ||||
| \(\mathfrak{D}\) | = | \((48828125,a + 18491818,a^{4} - a^{3} - 4 a^{2} + 2 a + 42643324,-a^{4} + a^{3} + 5 a^{2} - 3 a + 18029109,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 26244211)\) | = | \( \left(-a^{2} + a + 2\right)^{11} \) |
| sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| ||||
| \(N(\mathfrak{D})\) | = | \( 48828125 \) | = | \( 5^{11} \) |
| sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| ||||
| \(j\) | = | \( \frac{81829946}{3125} a^{4} - \frac{143808649}{3125} a^{3} - \frac{300612723}{3125} a^{2} + \frac{393132181}{3125} a + \frac{109010922}{3125} \) | ||
| sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| ||||
| \( \text{End} (E) \) | = | \(\Z\) | (no Complex Multiplication ) | |
| sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| ||||
| \( \text{ST} (E) \) | = | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
Rank not available.sage: E.rank()
magma: Rank(E);
Regulator: not available
sage: gens = E.gens(); gens
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
sage: E.regulator_of_points(gens)
magma: Regulator(gens);
Torsion subgroup
| Structure: | Trivial |
|---|---|
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a^{2} + a + 2\right) \) | \(5\) | \(4\) | \(I_{5}^*\) | Additive | \(1\) | \(2\) | \(11\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
| \(5\) | 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
25.1-c
consists of curves linked by isogenies of
degrees dividing 15.