Base field \(\Q(\sqrt{-47}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 12 \); class number \(5\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([12, -1, 1]))
gp: K = nfinit(Polrev([12, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([1,0]),K([12753,-2985]),K([-599175,-103117])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([1,0]),Polrev([12753,-2985]),Polrev([-599175,-103117])], K);
magma: E := EllipticCurve([K![0,0],K![0,-1],K![1,0],K![12753,-2985],K![-599175,-103117]]);
This is not a global minimal model: it is minimal at all primes except \((17,a+7)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((9,3a)\) | = | \((3,a)^{2}\cdot(3,a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27 \) | = | \(3^{2}\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2253088096392a+11368709749629)\) | = | \((3,a)^{23}\cdot(3,a+2)\cdot(17,a+7)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 164549728404324620728411041 \) | = | \(3^{23}\cdot3\cdot17^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((94143178827,3a+92428829235)\) | = | \((3,a)^{23}\cdot(3,a+2)\) |
Minimal discriminant norm: | \( 282429536481 \) | = | \(3^{23}\cdot3\) |
j-invariant: | \( \frac{1144804950016}{129140163} a - \frac{589982863360}{129140163} \) | ||
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: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 1.2899097428474887809290190565261049021 \) | ||
Tamagawa product: | \( 4 \) = \(2^{2}\cdot1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.5052213893880024312325017655039492183 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3,a)\) | \(3\) | \(4\) | \(I_{17}^{*}\) | Additive | \(-1\) | \(2\) | \(23\) | \(17\) |
\((3,a+2)\) | \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((17,a+7)\) | \(17\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(17\) | 17B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
17.
Its isogeny class
27.2-c
consists of curves linked by isogenies of
degree 17.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.