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Properties

Base field 6.6.485125.1
Label 6.6.485125.1-49.1-e4
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 \( 1 \)
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

\( y^2 + \left(a^{5} - 5 a^{3} + 4 a\right) x y + \left(-a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a + 1\right) y = x^{3} + \left(-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 7 a + 1\right) x^{2} + \left(-67 a^{5} + 147 a^{4} + 136 a^{3} - 479 a^{2} + 248 a - 44\right) x - 611 a^{5} + 1384 a^{4} + 1064 a^{3} - 4447 a^{2} + 2879 a - 492 \)
magma: E := ChangeRing(EllipticCurve([a^5 - 5*a^3 + 4*a, -a^5 + a^4 + 6*a^3 - 4*a^2 - 7*a + 1, -a^5 + a^4 + 5*a^3 - 3*a^2 - 5*a + 1, -67*a^5 + 147*a^4 + 136*a^3 - 479*a^2 + 248*a - 44, -611*a^5 + 1384*a^4 + 1064*a^3 - 4447*a^2 + 2879*a - 492]),K);
 
sage: E = EllipticCurve(K, [a^5 - 5*a^3 + 4*a, -a^5 + a^4 + 6*a^3 - 4*a^2 - 7*a + 1, -a^5 + a^4 + 5*a^3 - 3*a^2 - 5*a + 1, -67*a^5 + 147*a^4 + 136*a^3 - 479*a^2 + 248*a - 44, -611*a^5 + 1384*a^4 + 1064*a^3 - 4447*a^2 + 2879*a - 492])
 
gp (2.8): E = ellinit([a^5 - 5*a^3 + 4*a, -a^5 + a^4 + 6*a^3 - 4*a^2 - 7*a + 1, -a^5 + a^4 + 5*a^3 - 3*a^2 - 5*a + 1, -67*a^5 + 147*a^4 + 136*a^3 - 479*a^2 + 248*a - 44, -611*a^5 + 1384*a^4 + 1064*a^3 - 4447*a^2 + 2879*a - 492],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)
 
\(j\) = \( \frac{1868725201871035180068}{2401} a^{5} - \frac{49786647890889419082432}{16807} a^{4} + \frac{37590650528569269107756}{16807} a^{3} + \frac{36759694234274090980991}{16807} a^{2} - \frac{40226057957192491342390}{16807} a + \frac{7243098992357634858294}{16807} \)
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: Trivial
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]
 

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
\(3\) 3B
\(5\) 5B.1.4[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 49.1-e consists of curves linked by isogenies of degrees dividing 15.

Base change

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