Base field \(\Q(\sqrt{2}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 1]))
gp: K = nfinit(Polrev([-2, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([1,-1]),K([0,1]),K([-4,-3]),K([292,-213])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,-1]),Polrev([0,1]),Polrev([-4,-3]),Polrev([292,-213])], K);
magma: E := EllipticCurve([K![1,1],K![1,-1],K![0,1],K![-4,-3],K![292,-213]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((21a)\) | = | \((a)\cdot(-2a+1)\cdot(2a+1)\cdot(3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 882 \) | = | \(2\cdot7\cdot7\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((515403a-41958)\) | = | \((a)\cdot(-2a+1)\cdot(2a+1)^{8}\cdot(3)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -529520031054 \) | = | \(-2\cdot7\cdot7^{8}\cdot9^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{127949540041}{34588806} a + \frac{2442796570097}{466948881} \) | ||
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 | $\left(-11 a + 19 : 72 a - 102 : 1\right)$ |
| Height | \(0.13045176824092899273266601008012426414\) |
| 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: | $\left(\frac{5}{2} a - \frac{23}{4} : \frac{9}{8} a + \frac{3}{8} : 1\right)$ |
| 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: | \( 0.13045176824092899273266601008012426414 \) | ||
| Period: | \( 4.3090736250735678467920346328380627919 \) | ||
| Tamagawa product: | \( 16 \) = \(1\cdot1\cdot2^{3}\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.5899332005497986378925940077662732221 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((-2a+1)\) | \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((2a+1)\) | \(7\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((3)\) | \(9\) | \(2\) | \(I_{4}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
882.1-f
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.