Base field \(\Q(\sqrt{105}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 26 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-26, -1, 1]))
gp: K = nfinit(Polrev([-26, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([-337127,-72917]),K([112016739,24227822])])
gp: E = ellinit([Polrev([0,1]),Polrev([-1,-1]),Polrev([0,1]),Polrev([-337127,-72917]),Polrev([112016739,24227822])], K);
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![-337127,-72917],K![112016739,24227822]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((4a+19)\) | = | \((3,a+1)\cdot(7,a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2793a+18522)\) | = | \((2,a)^{12}\cdot(3,a+1)^{2}\cdot(7,a+3)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 88510464 \) | = | \(2^{12}\cdot3^{2}\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((147)\) | = | \((3,a+1)^{2}\cdot(7,a+3)^{4}\) |
Minimal discriminant norm: | \( 21609 \) | = | \(3^{2}\cdot7^{4}\) |
j-invariant: | \( \frac{53297461115137}{147} \) | ||
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: | \(1\) |
Generator | $\left(\frac{165705}{3388} a + \frac{54509}{242} : -\frac{6228155}{47432} a - \frac{158304087}{260876} : 1\right)$ |
Height | \(7.7259292850607236731371979260590027996\) |
Torsion structure: | \(\Z/4\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(48 a + 221 : -164 a - 758 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 7.7259292850607236731371979260590027996 \) | ||
Period: | \( 14.607527768551696889610408346184048628 \) | ||
Tamagawa product: | \( 4 \) = \(1\cdot2\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 5.5068443845900806297686718371416794056 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
\((3,a+1)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((7,a+3)\) | \(7\) | \(2\) | \(I_{4}\) | 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, 4 and 8.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 63.a1 |
\(\Q\) | 3675.n1 |