Base field \(\Q(\sqrt{205}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 51 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-51, -1, 1]))
gp: K = nfinit(Polrev([-51, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,0]),K([1,1]),K([179,30]),K([1275,204])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,0]),Polrev([1,1]),Polrev([179,30]),Polrev([1275,204])], K);
magma: E := EllipticCurve([K![1,1],K![-1,0],K![1,1],K![179,30],K![1275,204]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((14,2a+2)\) | = | \((2)\cdot(7,a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 28 \) | = | \(4\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-4267374a-30519320)\) | = | \((2)\cdot(7,a+1)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 132931722278404 \) | = | \(4\cdot7^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{106286015625663357}{66465861139202} a - \frac{353222055694778922}{33232930569601} \) | ||
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 | $\left(-\frac{3}{5} a + \frac{3}{5} : -\frac{89}{25} a + \frac{107}{25} : 1\right)$ | $\left(\frac{133}{9} a + \frac{1873}{15} : \frac{120521}{675} a + \frac{319294}{225} : 1\right)$ |
Heights | \(1.3276645659041626300030272492465037277\) | \(4.2804945973243316576482490358724199157\) |
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: | $\left(-\frac{3}{4} a - 2 : \frac{5}{4} a + \frac{157}{8} : 1\right)$ | |
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: | \( 5.4587435659360312087544588253619862003 \) | ||
Period: | \( 0.98580328583673249424639190347602297057 \) | ||
Tamagawa product: | \( 16 \) = \(1\cdot2^{4}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 6.0134818783477308128885275361820399861 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(4\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((7,a+1)\) | \(7\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
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
28.1-b
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.