Base field \(\Q(\sqrt{85}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 21 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-21, -1, 1]))
gp: K = nfinit(Polrev([-21, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,1]),K([-29,5]),K([-3646,-834])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,1]),Polrev([-29,5]),Polrev([-3646,-834])], K);
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,1],K![-29,5],K![-3646,-834]]);
This is not a global minimal model: it is minimal at all primes except \((3,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((15,3a+6)\) | = | \((3,a)\cdot(3,a+2)\cdot(5,a+2)\) |
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: | \((933075a-40556025)\) | = | \((3,a)^{24}\cdot(3,a+2)^{2}\cdot(5,a+2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -1588666142705625 \) | = | \(-3^{24}\cdot3^{2}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((13725a+24975)\) | = | \((3,a)^{12}\cdot(3,a+2)^{2}\cdot(5,a+2)^{4}\) |
Minimal discriminant norm: | \( -2989355625 \) | = | \(-3^{12}\cdot3^{2}\cdot5^{4}\) |
j-invariant: | \( -\frac{448290634633}{13286025} a + \frac{2290662945028}{13286025} \) | ||
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 | $\left(5 a + 15 : -27 a - 107 : 1\right)$ |
Height | \(0.99694558705177776320465040781943112024\) |
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: | $\left(3 a + \frac{27}{4} : -2 a - \frac{31}{8} : 1\right)$ |
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: | \( 0.99694558705177776320465040781943112024 \) | ||
Period: | \( 2.8548416648224762183522414008927478545 \) | ||
Tamagawa product: | \( 16 \) = \(2\cdot2\cdot2^{2}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.4696420198722209114224515089328893430 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3,a)\) | \(3\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
\((3,a+2)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
\((5,a+2)\) | \(5\) | \(4\) | \(I_{4}\) | 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 and 4.
Its isogeny class
45.1-e
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.