Properties

Base field 6.6.1241125.1
Label 6.6.1241125.1-25.2-b1
Conductor \((5,-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)\)
Conductor norm \( 25 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Base field 6.6.1241125.1

Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 7*x^4 - 2*x^3 + 11*x^2 + 7*x + 1)
 
gp: K = nfinit(a^6 - 7*a^4 - 2*a^3 + 11*a^2 + 7*a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
 

Weierstrass equation

\( y^2 + \left(-a^{5} + a^{4} + 6 a^{3} - 3 a^{2} - 8 a - 3\right) x y + \left(-3 a^{5} + a^{4} + 20 a^{3} - 29 a - 12\right) y = x^{3} + \left(-6 a^{5} + 2 a^{4} + 41 a^{3} - a^{2} - 65 a - 23\right) x^{2} + \left(-45 a^{5} + 22 a^{4} + 303 a^{3} - 59 a^{2} - 463 a - 84\right) x - 5610 a^{5} + 2584 a^{4} + 38079 a^{3} - 6320 a^{2} - 58796 a - 12189 \)
sage: E = EllipticCurve(K, [-a^5 + a^4 + 6*a^3 - 3*a^2 - 8*a - 3, -6*a^5 + 2*a^4 + 41*a^3 - a^2 - 65*a - 23, -3*a^5 + a^4 + 20*a^3 - 29*a - 12, -45*a^5 + 22*a^4 + 303*a^3 - 59*a^2 - 463*a - 84, -5610*a^5 + 2584*a^4 + 38079*a^3 - 6320*a^2 - 58796*a - 12189])
 
gp: E = ellinit([-a^5 + a^4 + 6*a^3 - 3*a^2 - 8*a - 3, -6*a^5 + 2*a^4 + 41*a^3 - a^2 - 65*a - 23, -3*a^5 + a^4 + 20*a^3 - 29*a - 12, -45*a^5 + 22*a^4 + 303*a^3 - 59*a^2 - 463*a - 84, -5610*a^5 + 2584*a^4 + 38079*a^3 - 6320*a^2 - 58796*a - 12189],K)
 
magma: E := ChangeRing(EllipticCurve([-a^5 + a^4 + 6*a^3 - 3*a^2 - 8*a - 3, -6*a^5 + 2*a^4 + 41*a^3 - a^2 - 65*a - 23, -3*a^5 + a^4 + 20*a^3 - 29*a - 12, -45*a^5 + 22*a^4 + 303*a^3 - 59*a^2 - 463*a - 84, -5610*a^5 + 2584*a^4 + 38079*a^3 - 6320*a^2 - 58796*a - 12189]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((5,-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)\) = \( \left(5, a - 2\right)^{2} \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 25 \) = \( 5^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((48828125,9765625 a^{5} - 68359375 a^{3} - 9765625 a^{2} + 107421875 a + 58593750,3476741 a^{5} + a^{4} - 24337188 a^{3} - 3476745 a^{2} + 38244154 a + 22385893,8316565 a^{5} + a^{4} - 58215955 a^{3} - 8316569 a^{2} + 91482214 a + 25896817,8522703 a^{5} - 59658921 a^{3} - 8522703 a^{2} + 93749734 a + 29165011,947122 a^{5} - 6629854 a^{3} - 947121 a^{2} + 10418342 a + 12105053)\) = \( \left(5, a - 2\right)^{21} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 476837158203125 \) = \( 5^{21} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{584205938671037576}{390625} a^{5} + \frac{127830640643771423}{390625} a^{4} + \frac{4061470830963324523}{390625} a^{3} + \frac{279717696988069228}{390625} a^{2} - \frac{1297494123233689326}{78125} a - \frac{2669912142068228202}{390625} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

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\((j)_{-}\)
\( \left(5, a - 2\right) \) \(5\) \(2\) \(I_{15}^*\) Additive \(1\) \(2\) \(21\) \(15\)

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
\(5\) 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 25.2-b 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.