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Properties

Base field 6.6.485125.1
Label 6.6.485125.1-49.1-c1
Conductor \((7,-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6)\)
Conductor norm \( 49 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field 6.6.485125.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 8*x^3 + 2*x^2 - 5*x + 1)
 
gp (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 8*a^3 + 2*a^2 - 5*a + 1);
 

Weierstrass equation

\( y^2 + \left(-a^{5} + 2 a^{4} + 5 a^{3} - 7 a^{2} - 6 a + 3\right) x y + \left(a^{5} - 5 a^{3} + 4 a + 1\right) y = x^{3} + \left(3 a^{5} - 4 a^{4} - 13 a^{3} + 13 a^{2} + 11 a - 5\right) x^{2} + \left(4 a^{5} - 7 a^{4} - 19 a^{3} + 24 a^{2} + 23 a - 7\right) x + 3 a^{5} - 3 a^{4} - 15 a^{3} + 11 a^{2} + 14 a - 5 \)
magma: E := ChangeRing(EllipticCurve([-a^5 + 2*a^4 + 5*a^3 - 7*a^2 - 6*a + 3, 3*a^5 - 4*a^4 - 13*a^3 + 13*a^2 + 11*a - 5, a^5 - 5*a^3 + 4*a + 1, 4*a^5 - 7*a^4 - 19*a^3 + 24*a^2 + 23*a - 7, 3*a^5 - 3*a^4 - 15*a^3 + 11*a^2 + 14*a - 5]),K);
 
sage: E = EllipticCurve(K, [-a^5 + 2*a^4 + 5*a^3 - 7*a^2 - 6*a + 3, 3*a^5 - 4*a^4 - 13*a^3 + 13*a^2 + 11*a - 5, a^5 - 5*a^3 + 4*a + 1, 4*a^5 - 7*a^4 - 19*a^3 + 24*a^2 + 23*a - 7, 3*a^5 - 3*a^4 - 15*a^3 + 11*a^2 + 14*a - 5])
 
gp (2.8): E = ellinit([-a^5 + 2*a^4 + 5*a^3 - 7*a^2 - 6*a + 3, 3*a^5 - 4*a^4 - 13*a^3 + 13*a^2 + 11*a - 5, a^5 - 5*a^3 + 4*a + 1, 4*a^5 - 7*a^4 - 19*a^3 + 24*a^2 + 23*a - 7, 3*a^5 - 3*a^4 - 15*a^3 + 11*a^2 + 14*a - 5],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((7,-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6)\) = \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 49 \) = \( 49 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((16807,16807 a^{5} - 16807 a^{4} - 84035 a^{3} + 67228 a^{2} + 84035 a - 33614,9583 a^{5} - 9582 a^{4} - 47916 a^{3} + 38329 a^{2} + 47917 a - 2467,14572 a^{5} - 14572 a^{4} - 72859 a^{3} + 58287 a^{2} + 72857 a - 19766,7141 a^{5} - 7141 a^{4} - 35705 a^{3} + 28564 a^{2} + 35706 a + 289,16616 a^{5} - 16616 a^{4} - 83080 a^{3} + 66465 a^{2} + 83080 a - 25885)\) = \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right)^{5} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 282475249 \) = \( 49^{5} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{795540805}{16807} a^{5} - \frac{1001826365}{16807} a^{4} - \frac{4000609051}{16807} a^{3} + \frac{3577994149}{16807} a^{2} + \frac{4448111321}{16807} a - \frac{1220726677}{16807} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.
magma: Rank(E);
 
sage: E.rank()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: not available

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generator: $\left(-a^{5} + a^{4} + 5 a^{3} - 4 a^{2} - 5 a + 2 : -a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[3]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right) \) \(49\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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.
Its isogeny class 49.1-c consists of curves linked by isogenies of degree 2.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.