Base field 4.4.9248.1
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([2, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,0]),K([4,-1,-1,0]),K([0,1,0,0]),K([581,870,-130,-192]),K([578,879,-121,-190])])
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([4,-1,-1,0]),Polrev([0,1,0,0]),Polrev([581,870,-130,-192]),Polrev([578,879,-121,-190])], K);
magma: E := EllipticCurve([K![0,1,0,0],K![4,-1,-1,0],K![0,1,0,0],K![581,870,-130,-192],K![578,879,-121,-190]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-4a-2)\) | = | \((a)^{3}\cdot(-a^3+4a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(2^{3}\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((10a^3-44a)\) | = | \((a)^{11}\cdot(-a^3+4a+1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 8192 \) | = | \(2^{11}\cdot2^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{246832014177796131}{2} a^{3} - 81720334491844608 a^{2} - 562968634281984056 a + 372771621664981728 \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{3}{4} a^{2} + a - 2 : -\frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{2} a : 1\right)$ |
| 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: | \( 41.861255705270412160590340629648579834 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.870599346417081 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(3\) | \(11\) | \(0\) |
| \((-a^3+4a+1)\) | \(2\) | \(2\) | \(I_{2}\) | Non-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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
16.2-d
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.