Base field 4.4.2225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 2 x + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([4, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-3/2,-1/2,1/2]),K([3,1,-1,0]),K([-2,0,1,0]),K([21,5/2,-51/2,17/2]),K([76,-3/2,-189/2,67/2])])
gp: E = ellinit([Polrev([0,-3/2,-1/2,1/2]),Polrev([3,1,-1,0]),Polrev([-2,0,1,0]),Polrev([21,5/2,-51/2,17/2]),Polrev([76,-3/2,-189/2,67/2])], K);
magma: E := EllipticCurve([K![0,-3/2,-1/2,1/2],K![3,1,-1,0],K![-2,0,1,0],K![21,5/2,-51/2,17/2],K![76,-3/2,-189/2,67/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^3-3/2a^2-1/2a+4)\) | = | \((1/2a^3-3/2a^2-1/2a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((13/2a^3-31/2a^2-29/2a+47)\) | = | \((1/2a^3-3/2a^2-1/2a+4)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 130321 \) | = | \(19^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{6593184883}{130321} a^{3} + \frac{2678496927}{130321} a^{2} + \frac{26352219634}{130321} a + \frac{16630098173}{130321} \) | ||
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: | \(0\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(a^{3} - 2 a^{2} - 2 a + 2 : \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{1}{2} a + 1 : 1\right)$ | $\left(\frac{3}{8} a^{3} - \frac{7}{8} a^{2} + \frac{1}{8} a - \frac{1}{4} : \frac{1}{2} a^{3} - \frac{13}{8} a^{2} - \frac{5}{8} a + \frac{15}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 434.85492064667811289629217138118454955 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.15236323498953 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-3/2a^2-1/2a+4)\) | \(19\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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
19.1-a
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.