Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 5*x^3 - x^2 + 3*x + 1)
gp (2.8): K = nfinit(a^5 - 5*a^3 - a^2 + 3*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^4 - a^3 - 4*a^2 + 4*a + 2, -3*a^4 + a^3 + 14*a^2 - 3*a - 6, a^4 - a^3 - 4*a^2 + 4*a + 2, -50*a^4 + 26*a^3 + 232*a^2 - 74*a - 103, 65*a^4 - 10*a^3 - 330*a^2 - 4*a + 225]),K);
sage: E = EllipticCurve(K, [a^4 - a^3 - 4*a^2 + 4*a + 2, -3*a^4 + a^3 + 14*a^2 - 3*a - 6, a^4 - a^3 - 4*a^2 + 4*a + 2, -50*a^4 + 26*a^3 + 232*a^2 - 74*a - 103, 65*a^4 - 10*a^3 - 330*a^2 - 4*a + 225])
gp (2.8): E = ellinit([a^4 - a^3 - 4*a^2 + 4*a + 2, -3*a^4 + a^3 + 14*a^2 - 3*a - 6, a^4 - a^3 - 4*a^2 + 4*a + 2, -50*a^4 + 26*a^3 + 232*a^2 - 74*a - 103, 65*a^4 - 10*a^3 - 330*a^2 - 4*a + 225],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((85,a^{4} - 5 a^{2} - 2 a + 3)\) | = | \( \left(2 a^{4} - a^{3} - 9 a^{2} + 2 a + 3\right) \cdot \left(a^{2} - a - 3\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 85 \) | = | \( 5 \cdot 17 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((1445,a^{4} - 5 a^{2} + 79,-a^{4} + 5 a^{2} + a + 554,2 a^{4} - a^{3} - 9 a^{2} + 3 a + 566,-a^{4} + a^{3} + 5 a^{2} - 3 a + 534)\) | = | \( \left(2 a^{4} - a^{3} - 9 a^{2} + 2 a + 3\right) \cdot \left(a^{2} - a - 3\right)^{2} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 1445 \) | = | \( 5 \cdot 17^{2} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{185066934807448}{1445} a^{4} - \frac{87464764407119}{1445} a^{3} - \frac{909727616081443}{1445} a^{2} + \frac{231799410298786}{1445} a + \frac{559295724774971}{1445} \) | ||
| 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: | $\left(\frac{15}{4} a^{4} - \frac{3}{2} a^{3} - \frac{71}{4} a^{2} + \frac{15}{4} a + \frac{35}{4} : \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{13}{8} a^{2} + \frac{1}{2} a + \frac{1}{4} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(2 a^{4} - a^{3} - 9 a^{2} + 2 a + 3\right) \) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \( \left(a^{2} - a - 3\right) \) | \(17\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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.
Its isogeny class
85.2-b
consists of curves linked by isogenies of
degree 2.