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([1,0]),K([0,0]),K([1,0]),K([-23211,-14480]),K([-1786730,-1224480])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-23211,-14480]),Polrev([-1786730,-1224480])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-23211,-14480],K![-1786730,-1224480]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((7a)\) | = | \((a)\cdot(-2a+1)\cdot(2a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 98 \) | = | \(2\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((2128a+672)\) | = | \((a)^{9}\cdot(-2a+1)\cdot(2a+1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 8605184 \) | = | \(2^{9}\cdot7\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{218768831290078842857759125}{76832} a + \frac{9668307757341907702465000}{2401} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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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: | $\left(-64 a - \frac{121}{4} : 32 a + \frac{117}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 0.21809533038171905844615354057497739059 \) | ||
Tamagawa product: | \( 4 \) = \(1\cdot1\cdot2^{2}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 0.69397509176115303883636998319217787197 \) | ||
Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(1\) | \(I_{9}\) | Non-split multiplicative | \(1\) | \(1\) | \(9\) | \(9\) |
\((-2a+1)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((2a+1)\) | \(7\) | \(4\) | \(I_{4}\) | 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 |
\(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6, 9, 12, 18 and 36.
Its isogeny class
98.1-a
consists of curves linked by isogenies of
degrees dividing 36.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.