Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 8*x^3 + 2*x^2 - 5*x + 1)
gp (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 8*a^3 + 2*a^2 - 5*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([-a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 3, a^5 - a^4 - 5*a^3 + 4*a^2 + 4*a - 1, a^5 - a^4 - 5*a^3 + 4*a^2 + 6*a - 1, a^5 - 4*a^4 - 8*a^3 + 14*a^2 + 9*a - 10, 6*a^5 - 4*a^4 - 24*a^3 + 17*a^2 + 13*a - 14]),K);
sage: E = EllipticCurve(K, [-a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 3, a^5 - a^4 - 5*a^3 + 4*a^2 + 4*a - 1, a^5 - a^4 - 5*a^3 + 4*a^2 + 6*a - 1, a^5 - 4*a^4 - 8*a^3 + 14*a^2 + 9*a - 10, 6*a^5 - 4*a^4 - 24*a^3 + 17*a^2 + 13*a - 14])
gp (2.8): E = ellinit([-a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 3, a^5 - a^4 - 5*a^3 + 4*a^2 + 4*a - 1, a^5 - a^4 - 5*a^3 + 4*a^2 + 6*a - 1, a^5 - 4*a^4 - 8*a^3 + 14*a^2 + 9*a - 10, 6*a^5 - 4*a^4 - 24*a^3 + 17*a^2 + 13*a - 14],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((7,-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6)\) | = | \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 49 \) | = | \( 49 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((343,343 a^{5} - 343 a^{4} - 1715 a^{3} + 1372 a^{2} + 1715 a - 686,322 a^{5} - 321 a^{4} - 1611 a^{3} + 1285 a^{2} + 1612 a - 409,166 a^{5} - 166 a^{4} - 829 a^{3} + 663 a^{2} + 827 a - 215,281 a^{5} - 281 a^{4} - 1405 a^{3} + 1124 a^{2} + 1406 a - 397,152 a^{5} - 152 a^{4} - 760 a^{3} + 609 a^{2} + 760 a - 160)\) | = | \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right)^{3} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 117649 \) | = | \( 49^{3} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{30176477175473918504}{49} a^{5} - \frac{370959595583589700360}{343} a^{4} - \frac{935402224099909297679}{343} a^{3} + \frac{1461778923930007873681}{343} a^{2} + \frac{778934793586578494395}{343} a - \frac{866228484050474713990}{343} \) | ||
| 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: | Trivial |
|---|---|
| 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]
| |
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right) \) | \(49\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
49.1-b
consists of curves linked by isogenies of
degree 3.