Base field \(\Q(\sqrt{65}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 16 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-16, -1, 1]))
gp: K = nfinit(Polrev([-16, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([0,1]),K([1067,-241]),K([149,-29])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([0,1]),Polrev([1067,-241]),Polrev([149,-29])], K);
magma: E := EllipticCurve([K![1,0],K![-1,1],K![0,1],K![1067,-241],K![149,-29]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((14)\) | = | \((2,a)\cdot(2,a+1)\cdot(7,a+1)\cdot(7,a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 196 \) | = | \(2\cdot2\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-10449152a+71663872)\) | = | \((2,a)^{8}\cdot(2,a+1)^{32}\cdot(7,a+1)^{2}\cdot(7,a+5)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -2639927418290176 \) | = | \(-2^{8}\cdot2^{32}\cdot7^{2}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{10188791249212817}{210453397504} a + \frac{2271155657117313}{13153337344} \) | ||
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: | $\left(\frac{1}{4} a + \frac{3}{4} : -\frac{5}{8} a - \frac{3}{8} : 1\right)$ |
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: | \( 2.4247811975237109425321146979503134317 \) | ||
Tamagawa product: | \( 16 \) = \(2\cdot2\cdot2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.2030283690870261257678602149920100445 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((2,a+1)\) | \(2\) | \(2\) | \(I_{32}\) | Non-split multiplicative | \(1\) | \(1\) | \(32\) | \(32\) |
\((7,a+1)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((7,a+5)\) | \(7\) | \(2\) | \(I_{2}\) | 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
196.1-c
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.