Base field \(\Q(\sqrt{2}, \sqrt{5})\)
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 0, -6, 0, 1]))
gp: K = nfinit(Polrev([4, 0, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,1,1/2,0]),K([2,-1,-1/2,1/2]),K([1,-2,0,1/2]),K([75,-37,-199/2,45]),K([582,-256,-1537/2,673/2])])
gp: E = ellinit([Polrev([-1,1,1/2,0]),Polrev([2,-1,-1/2,1/2]),Polrev([1,-2,0,1/2]),Polrev([75,-37,-199/2,45]),Polrev([582,-256,-1537/2,673/2])], K);
magma: E := EllipticCurve([K![-1,1,1/2,0],K![2,-1,-1/2,1/2],K![1,-2,0,1/2],K![75,-37,-199/2,45],K![582,-256,-1537/2,673/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/2a^2+a+3)\) | = | \((-1/2a^2+a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 31 \) | = | \(31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((7561/2a^3+24553/2a^2-8230a-49762)\) | = | \((-1/2a^2+a+3)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -787662783788549761 \) | = | \(-31^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{18294256594913191872513}{787662783788549761} a^{3} - \frac{29872156558690483602305}{1575325567577099522} a^{2} + \frac{95382125225553480591493}{787662783788549761} a + \frac{80704205874773584791012}{787662783788549761} \) | ||
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: | $\left(\frac{5}{4} a^{3} - \frac{23}{8} a^{2} - \frac{5}{2} a + 2 : \frac{9}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{8} : 1\right)$ |
| 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: | \( 21.328917542675473882006677158657403896 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.06644587713377 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/2a^2+a+3)\) | \(31\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
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 |
| \(3\) | 3B |
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
31.3-a
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.