Properties

Label 6.6.1387029.1-21.1-b6
Base field 6.6.1387029.1
Conductor norm \( 21 \)
CM no
Base change yes
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 6.6.1387029.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
 
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-2a^{3}-2a^{2}+3a+1\right){x}{y}+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+4\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{5}-232a^{4}+455a^{3}+979a^{2}-1196a-1143\right){x}-3103a^{4}+6206a^{3}+13092a^{2}-16196a-15855\)
sage: E = EllipticCurve([K([1,3,-2,-2,1,0]),K([-2,-1,1,0,0,0]),K([4,-4,-13,6,5,-2]),K([-1143,-1196,979,455,-232,1]),K([-15855,-16196,13092,6206,-3103,0])])
 
gp: E = ellinit([Polrev([1,3,-2,-2,1,0]),Polrev([-2,-1,1,0,0,0]),Polrev([4,-4,-13,6,5,-2]),Polrev([-1143,-1196,979,455,-232,1]),Polrev([-15855,-16196,13092,6206,-3103,0])], K);
 
magma: E := EllipticCurve([K![1,3,-2,-2,1,0],K![-2,-1,1,0,0,0],K![4,-4,-13,6,5,-2],K![-1143,-1196,979,455,-232,1],K![-15855,-16196,13092,6206,-3103,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^5+5a^4+6a^3-14a^2-3a+4)\) = \((a^5-3a^4-2a^3+8a^2+a-2)\cdot(a^2-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 21 \) = \(3\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-16a^4+32a^3+57a^2-73a-28)\) = \((a^5-3a^4-2a^3+8a^2+a-2)^{4}\cdot(a^2-2)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 466948881 \) = \(3^{4}\cdot7^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2905242808362156164576920441}{21609} a^{4} - \frac{5810485616724312329153840882}{21609} a^{3} - \frac{12198217539758009162689930484}{21609} a^{2} + \frac{719212397529531682250802425}{1029} a + \frac{14621896551411174957571933555}{21609} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(-14 a^{5} + 23 a^{4} + 86 a^{3} - 97 a^{2} - 134 a + 91 : -36 a^{5} + 214 a^{4} - 309 a^{3} - 326 a^{2} + 952 a - 435 : 1\right)$
Height \(1.1731957583480136207825413586508381139\)
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: $\left(-\frac{9}{2} a^{4} + 9 a^{3} + \frac{69}{4} a^{2} - \frac{87}{4} a - \frac{33}{2} : a^{5} - \frac{1}{4} a^{4} - \frac{15}{2} a^{3} - \frac{29}{8} a^{2} + \frac{115}{8} a + \frac{101}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.1731957583480136207825413586508381139 \)
Period: \( 1.3086722316626959187897553071279438812 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 4.00479 \)
Analytic order of Ш: \( 256 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((a^5-3a^4-2a^3+8a^2+a-2)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((a^2-2)\) \(7\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

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, 4 and 8.
Its isogeny class 21.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
3.3.257.1 3.3.257.1-21.1-b6
3.3.257.1 a curve with conductor norm 441 (not in the database)