Base field 5.5.36497.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1)
gp (2.8): K = nfinit(a^5 - 2*a^4 - 3*a^3 + 5*a^2 + a - 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 - 2, a^4 - 2*a^3 - 3*a^2 + 5*a, a^4 - 2*a^3 - 3*a^2 + 6*a + 1, 21*a^4 + 48*a^3 - 214*a^2 - 40*a + 28, -319*a^4 + 696*a^3 + 155*a^2 - 141*a + 20]),K);
sage: E = EllipticCurve(K, [a^2 - 2, a^4 - 2*a^3 - 3*a^2 + 5*a, a^4 - 2*a^3 - 3*a^2 + 6*a + 1, 21*a^4 + 48*a^3 - 214*a^2 - 40*a + 28, -319*a^4 + 696*a^3 + 155*a^2 - 141*a + 20])
gp (2.8): E = ellinit([a^2 - 2, a^4 - 2*a^3 - 3*a^2 + 5*a, a^4 - 2*a^3 - 3*a^2 + 6*a + 1, 21*a^4 + 48*a^3 - 214*a^2 - 40*a + 28, -319*a^4 + 696*a^3 + 155*a^2 - 141*a + 20],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((39,2 a^{4} - 3 a^{3} - 7 a^{2} + 6 a + 2)\) | = | \( \left(a^{2} - 1\right) \cdot \left(a^{3} - 3 a\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 39 \) | = | \( 3 \cdot 13 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((28431,a + 18742,a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 20609,a^{4} - a^{3} - 4 a^{2} + 2 a + 15061,a^{2} - a + 12130)\) | = | \( \left(a^{2} - 1\right)^{7} \cdot \left(a^{3} - 3 a\right) \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 28431 \) | = | \( 3^{7} \cdot 13 \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{250577525153856331}{28431} a^{4} + \frac{23688817181716957}{9477} a^{3} - \frac{188375931275690315}{9477} a^{2} - \frac{51297605922912091}{28431} a + \frac{111644917373722610}{28431} \) | ||
| 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: | $\left(-\frac{13}{4} a^{4} + 4 a^{3} + 8 a^{2} - a + 1 : -\frac{3}{8} a^{4} + \frac{9}{8} a^{3} + \frac{9}{8} a^{2} - \frac{29}{8} a + \frac{7}{4} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} - 1\right) \) | \(3\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
| \( \left(a^{3} - 3 a\right) \) | \(13\) | \(1\) | \(I_{1}\) | 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 |
| \(7\) | 7B.6.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 7, 14 and 28.
Its isogeny class
39.1-a
consists of curves linked by isogenies of
degrees dividing 28.