Base field 3.1.23.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 + 1)
gp (2.8): K = nfinit(a^3 - a^2 + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 + 1, a - 1, 0, -449*a^2 + 838*a - 639, -6934*a^2 + 12248*a - 9210]),K);
sage: E = EllipticCurve(K, [a^2 + 1, a - 1, 0, -449*a^2 + 838*a - 639, -6934*a^2 + 12248*a - 9210])
gp (2.8): E = ellinit([a^2 + 1, a - 1, 0, -449*a^2 + 838*a - 639, -6934*a^2 + 12248*a - 9210],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((253,2 a^{2} - 7 a)\) | = | \( \left(a^{2} + a - 2\right) \cdot \left(-a^{2} - 2 a + 2\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 253 \) | = | \( 11 \cdot 23 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((1146551135499121,a + 487247944869930,a^{2} - a + 431803671072449)\) | = | \( \left(a^{2} + a - 2\right)^{4} \cdot \left(-a^{2} - 2 a + 2\right)^{8} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 1146551135499121 \) | = | \( 11^{4} \cdot 23^{8} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{833654495257217661469537}{1146551135499121} a^{2} - \frac{1463012109643929405934449}{1146551135499121} a + \frac{1104329696417202846546921}{1146551135499121} \) | ||
| 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: \( 0 \)magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
Regulator: 1
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: | $\left(-\frac{19}{4} a^{2} + \frac{45}{4} a - 11 : \frac{37}{8} a^{2} - 8 a + \frac{35}{4} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} + a - 2\right) \) | \(11\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \( \left(-a^{2} - 2 a + 2\right) \) | \(23\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
253.1-A
consists of curves linked by isogenies of
degrees dividing 8.