Base field 3.3.148.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 3*x + 1)
gp (2.8): K = nfinit(a^3 - a^2 - 3*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 - 1, a^2 - 3, 0, -43485*a^2 - 50880*a + 20040, -9168052*a^2 - 10727414*a + 4224740]),K);
sage: E = EllipticCurve(K, [a^2 - 1, a^2 - 3, 0, -43485*a^2 - 50880*a + 20040, -9168052*a^2 - 10727414*a + 4224740])
gp (2.8): E = ellinit([a^2 - 1, a^2 - 3, 0, -43485*a^2 - 50880*a + 20040, -9168052*a^2 - 10727414*a + 4224740],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((10,-a^{2} + 2 a + 3)\) | = | \( \left(a^{2} - a - 2\right)^{2} \cdot \left(a^{2} - a - 1\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 20 \) | = | \( 2^{2} \cdot 5 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((125000,8 a + 85736,4 a^{2} + 83644)\) | = | \( \left(a^{2} - a - 2\right)^{8} \cdot \left(a^{2} - a - 1\right)^{6} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 4000000 \) | = | \( 2^{8} \cdot 5^{6} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{7209817568672}{15625} a^{2} + \frac{8324376473504}{15625} a - \frac{3284389248384}{15625} \) | ||
| 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: | \(\Z/2\Z\times\Z/2\Z\) |
|---|---|
| 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]
| |
| Generators: | $\left(92 a^{2} + 108 a - 42 : -171 a^{2} - 200 a + 79 : 1\right)$,$\left(-\frac{93}{2} a^{2} - \frac{109}{2} a + 22 : 86 a^{2} + 101 a - \frac{79}{2} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} - a - 2\right) \) | \(2\) | \(1\) | \(IV^*\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \( \left(a^{2} - a - 1\right) \) | \(5\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
20.1-a
consists of curves linked by isogenies of
degrees dividing 24.