Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 6*x^2 - 3*x + 3)
gp (2.8): K = nfinit(a^4 - 6*a^2 - 3*a + 3);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([0, -a^3 + a^2 + 3*a, a + 1, -a^3 + 2*a^2 + 3*a - 1, -5*a^3 + 6*a^2 + 21*a - 12]),K);
sage: E = EllipticCurve(K, [0, -a^3 + a^2 + 3*a, a + 1, -a^3 + 2*a^2 + 3*a - 1, -5*a^3 + 6*a^2 + 21*a - 12])
gp (2.8): E = ellinit([0, -a^3 + a^2 + 3*a, a + 1, -a^3 + 2*a^2 + 3*a - 1, -5*a^3 + 6*a^2 + 21*a - 12],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((25,a^{3} - 2 a^{2} - 4 a + 5)\) | = | \( \left(-a^{3} + 2 a^{2} + 3 a - 4\right)^{2} \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 25 \) | = | \( 5^{2} \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((390625,a + 206579,a^{3} - a^{2} - 4 a + 193214,a^{2} - a + 300805)\) | = | \( \left(-a^{3} + 2 a^{2} + 3 a - 4\right)^{8} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 390625 \) | = | \( 5^{8} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( 0 \) | ||
| magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
| ||||
| \( \text{End} (E) \) | = | \(\Z[(1+\sqrt{-3})/2]\) | ( Complex Multiplication ) | |
| magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
| ||||
| \( \text{ST} (E) \) | = | $N(\mathrm{U}(1))$ | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a^{3} + 2 a^{2} + 3 a - 4\right) \) | \(5\) | \(1\) | \(IV^*\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.2 |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
25.1-e
consists of curves linked by isogenies of
degree 3.