Base field 4.4.9225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 10 x^{2} + 7 x + 19 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([19, 7, -10, -1, 1]))
gp: K = nfinit(Polrev([19, 7, -10, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 7, -10, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1/4,-3/2,0,1/4]),K([-5/4,1/2,0,-1/4]),K([-19/4,-3/2,1,1/4]),K([3/4,-3/2,-1,-1/4]),K([-20,-13,6,3])])
gp: E = ellinit([Polrev([1/4,-3/2,0,1/4]),Polrev([-5/4,1/2,0,-1/4]),Polrev([-19/4,-3/2,1,1/4]),Polrev([3/4,-3/2,-1,-1/4]),Polrev([-20,-13,6,3])], K);
magma: E := EllipticCurve([K![1/4,-3/2,0,1/4],K![-5/4,1/2,0,-1/4],K![-19/4,-3/2,1,1/4],K![3/4,-3/2,-1,-1/4],K![-20,-13,6,3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^3-3a-1/2)\) | = | \((1/2a^3-3a-1/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5)\) | = | \((1/2a^3-3a-1/2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 625 \) | = | \(25^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{157869}{5} a^{3} - \frac{328006}{5} a^{2} - \frac{1190227}{5} a + \frac{2483489}{5} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) | |
| Generator | $\left(\frac{3}{4} a^{3} + 3 a^{2} - \frac{7}{2} a - \frac{25}{4} : -10 a^{3} - 12 a^{2} + 47 a + 57 : 1\right)$ | |
| Height | \(1.4698381415082935216892734860569099387\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-\frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4} : \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{5}{8} a + \frac{5}{8} : 1\right)$ | $\left(\frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4} : -\frac{1}{4} a^{3} + \frac{5}{2} a + \frac{3}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 1.4698381415082935216892734860569099387 \) | ||
| Period: | \( 1324.9430586788292590262857837530332727 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 2.53450733678301 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-3a-1/2)\) | \(25\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
25.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.