Base field 4.4.725.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 3 x^{2} + x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 3*x^2 + x + 1)
gp (2.8): K = nfinit(a^4 - a^3 - 3*a^2 + a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^3 - a^2 - 2*a + 1, a^3 - 4*a + 1, a^3 - a^2 - 2*a + 2, 237*a^3 - 560*a^2 + 85*a + 103, 4228*a^3 - 10047*a^2 + 1135*a + 2839]),K);
sage: E = EllipticCurve(K, [a^3 - a^2 - 2*a + 1, a^3 - 4*a + 1, a^3 - a^2 - 2*a + 2, 237*a^3 - 560*a^2 + 85*a + 103, 4228*a^3 - 10047*a^2 + 1135*a + 2839])
gp (2.8): E = ellinit([a^3 - a^2 - 2*a + 1, a^3 - 4*a + 1, a^3 - a^2 - 2*a + 2, 237*a^3 - 560*a^2 + 85*a + 103, 4228*a^3 - 10047*a^2 + 1135*a + 2839],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((7,a^{2} + a - 3)\) | = | \( \left(a^{2} + a - 3\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 49 \) | = | \( 49 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((16807,16807 a^{3} - 16807 a^{2} - 33614 a + 16807,15150 a^{3} - 15150 a^{2} - 30299 a + 17211,2061 a^{3} - 2060 a^{2} - 4123 a + 15149)\) | = | \( \left(a^{2} + a - 3\right)^{5} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 282475249 \) | = | \( 49^{5} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{252950065091819511}{16807} a^{3} - \frac{373671346411971114}{16807} a^{2} - \frac{82930605545918026}{2401} a + \frac{530004191810724000}{16807} \) | ||
| 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: | $\left(-a^{3} - 5 a^{2} + 14 a + 1 : 2 a^{3} - 5 a^{2} - a + 5 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} + a - 3\right) \) | \(49\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(5\) | 5B.1.4[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
49.2-a
consists of curves linked by isogenies of
degrees dividing 10.