Base field \(\Q(\sqrt{105}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 26 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-26, -1, 1]))
gp: K = nfinit(Polrev([-26, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,1]),K([-56473,-12214]),K([8025336,1735781])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([0,1]),Polrev([-56473,-12214]),Polrev([8025336,1735781])], K);
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,1],K![-56473,-12214],K![8025336,1735781]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((35,a+17)\) | = | \((2a-11)\cdot(7,a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 35 \) | = | \(5\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((259765625a-1722656250)\) | = | \((2,a)^{12}\cdot(2a-11)^{18}\cdot(7,a+3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 765625000000000000 \) | = | \(2^{12}\cdot5^{18}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((13671875)\) | = | \((2a-11)^{18}\cdot(7,a+3)^{2}\) |
Minimal discriminant norm: | \( 186920166015625 \) | = | \(5^{18}\cdot7^{2}\) |
j-invariant: | \( -\frac{250523582464}{13671875} \) | ||
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: | \(\Z/3\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(12 a + \frac{167}{3} : -\frac{1817}{9} a - \frac{8375}{9} : 1\right)$ |
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: | \( 4.8622202595689165453532247611092932944 \) | ||
Tamagawa product: | \( 36 \) = \(1\cdot( 2 \cdot 3^{2} )\cdot2\) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 1.8980164424020566348409636192619640398 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
\((2a-11)\) | \(5\) | \(18\) | \(I_{18}\) | Split multiplicative | \(-1\) | \(1\) | \(18\) | \(18\) |
\((7,a+3)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
35.1-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 315.b1 |
\(\Q\) | 1225.e1 |