Base field \(\Q(\sqrt{34}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 34 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-34, 0, 1]))
gp: K = nfinit(Polrev([-34, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-34, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([1,0]),K([-146848875,-25184377]),K([940851899017,161354769508])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,-1]),Polrev([1,0]),Polrev([-146848875,-25184377]),Polrev([940851899017,161354769508])], K);
magma: E := EllipticCurve([K![1,0],K![1,-1],K![1,0],K![-146848875,-25184377],K![940851899017,161354769508]]);
This is not a global minimal model: it is minimal at all primes except \((3,a+1)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-3a-18)\) | = | \((-a-6)\cdot(3,a+1)\cdot(3,a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 18 \) | = | \(2\cdot3\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-767092568220a-6771335306742)\) | = | \((-a-6)^{2}\cdot(3,a+1)^{32}\cdot(3,a+2)^{20}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 25844327556906693195728964 \) | = | \(2^{2}\cdot3^{32}\cdot3^{20}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((6973568802)\) | = | \((-a-6)^{2}\cdot(3,a+1)^{20}\cdot(3,a+2)^{20}\) |
Minimal discriminant norm: | \( 48630661836227715204 \) | = | \(2^{2}\cdot3^{20}\cdot3^{20}\) |
j-invariant: | \( \frac{211293405175481}{6973568802} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(-1691 a - \frac{39453}{4} : \frac{1691}{2} a + \frac{39449}{8} : 1\right)$ | $\left(728 a + 4243 : -364 a - 2122 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 1.7437594210748084201039591294847480380 \) | ||
Tamagawa product: | \( 80 \) = \(2\cdot2\cdot( 2^{2} \cdot 5 )\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 2.9905227354324995095935599827415531646 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a-6)\) | \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((3,a+1)\) | \(3\) | \(2\) | \(I_{20}\) | Non-split multiplicative | \(1\) | \(1\) | \(20\) | \(20\) |
\((3,a+2)\) | \(3\) | \(20\) | \(I_{20}\) | Split multiplicative | \(-1\) | \(1\) | \(20\) | \(20\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
\(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
18.1-a
consists of curves linked by isogenies of
degrees dividing 20.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.