Base field \(\Q(\sqrt{26}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 26 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-26, 0, 1]))
gp: K = nfinit(Polrev([-26, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([0,0]),K([-382,76]),K([7600,-1490])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,1]),Polrev([0,0]),Polrev([-382,76]),Polrev([7600,-1490])], K);
magma: E := EllipticCurve([K![0,0],K![-1,1],K![0,0],K![-382,76],K![7600,-1490]]);
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: | \((3)\) | = | \((3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-46656)\) | = | \((2,a)^{12}\cdot(3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 2176782336 \) | = | \(2^{12}\cdot9^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((729)\) | = | \((3)^{6}\) |
Minimal discriminant norm: | \( 531441 \) | = | \(9^{6}\) |
j-invariant: | \( -\frac{778688}{729} \) | ||
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: | \(2\) | |
Generators | $\left(-\frac{87}{25} a + \frac{412}{25} : \frac{1809}{125} a - \frac{9234}{125} : 1\right)$ | $\left(-\frac{33}{25} a + \frac{142}{25} : \frac{999}{125} a - \frac{5076}{125} : 1\right)$ |
Heights | \(0.57246970603315629727170505178953459146\) | \(0.57246970603315629727170505178953459146\) |
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);
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Torsion generator: | $\left(3 a - 17 : 0 : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.32772156432568838750346252628210895876 \) | ||
Period: | \( 21.409495403152498809394153474764730497 \) | ||
Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.3760200968521156620665826238903668220 \) | ||
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 | \(1\) | \(0\) | \(0\) | \(0\) |
\((3)\) | \(9\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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 |
\(3\) | 3Nn |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
9.1-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.