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([1,0]),K([1,1]),K([1,1]),K([-14369,1879]),K([606185522,-79147747])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,1]),Polrev([1,1]),Polrev([-14369,1879]),Polrev([606185522,-79147747])], K);
magma: E := EllipticCurve([K![1,0],K![1,1],K![1,1],K![-14369,1879],K![606185522,-79147747]]);
This is not a global minimal model: it is minimal at all primes except \((3,a+2)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((3a+21)\) | = | \((3,a)\cdot(3,a+2)\cdot(a+7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 45 \) | = | \(3\cdot3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((252899485875a-2099603816775)\) | = | \((3,a)^{16}\cdot(3,a+2)^{28}\cdot(a+7)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 615481813864757020550625 \) | = | \(3^{16}\cdot3^{28}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((1076168025)\) | = | \((3,a)^{16}\cdot(3,a+2)^{16}\cdot(a+7)^{4}\) |
Minimal discriminant norm: | \( 1158137618032400625 \) | = | \(3^{16}\cdot3^{16}\cdot5^{4}\) |
j-invariant: | \( -\frac{4173281}{1076168025} \) | ||
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 | $\left(-1061 a + 8124 : -130204 a + 997219 : 1\right)$ |
Height | \(0.93995547054071209529472531298800060121\) |
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: | $\left(73 a - \frac{2253}{4} : -37 a + \frac{2249}{8} : 1\right)$ |
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: | \( 0.93995547054071209529472531298800060121 \) | ||
Period: | \( 1.5275148272821564422777163613041010276 \) | ||
Tamagawa product: | \( 64 \) = \(2\cdot2^{4}\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.2089707770824399481203613811671351180 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3,a)\) | \(3\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
\((3,a+2)\) | \(3\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
\((a+7)\) | \(5\) | \(2\) | \(I_{4}\) | 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.
Its isogeny class
45.1-a
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.