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Properties

Base field 6.6.485125.1
Label 6.6.485125.1-49.1-e3
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

\( y^2 + \left(a^{4} - 4 a^{2} - a + 2\right) x y + \left(-2 a^{5} + 3 a^{4} + 9 a^{3} - 10 a^{2} - 7 a + 3\right) y = x^{3} + \left(a^{4} - 4 a^{2} + 2\right) x^{2} + \left(1286 a^{5} - 1768 a^{4} - 6218 a^{3} + 6370 a^{2} + 6394 a - 2350\right) x - 41543 a^{5} + 53994 a^{4} + 204324 a^{3} - 189619 a^{2} - 217635 a + 56419 \)
magma: E := ChangeRing(EllipticCurve([a^4 - 4*a^2 - a + 2, a^4 - 4*a^2 + 2, -2*a^5 + 3*a^4 + 9*a^3 - 10*a^2 - 7*a + 3, 1286*a^5 - 1768*a^4 - 6218*a^3 + 6370*a^2 + 6394*a - 2350, -41543*a^5 + 53994*a^4 + 204324*a^3 - 189619*a^2 - 217635*a + 56419]),K);
 
sage: E = EllipticCurve(K, [a^4 - 4*a^2 - a + 2, a^4 - 4*a^2 + 2, -2*a^5 + 3*a^4 + 9*a^3 - 10*a^2 - 7*a + 3, 1286*a^5 - 1768*a^4 - 6218*a^3 + 6370*a^2 + 6394*a - 2350, -41543*a^5 + 53994*a^4 + 204324*a^3 - 189619*a^2 - 217635*a + 56419])
 
gp (2.8): E = ellinit([a^4 - 4*a^2 - a + 2, a^4 - 4*a^2 + 2, -2*a^5 + 3*a^4 + 9*a^3 - 10*a^2 - 7*a + 3, 1286*a^5 - 1768*a^4 - 6218*a^3 + 6370*a^2 + 6394*a - 2350, -41543*a^5 + 53994*a^4 + 204324*a^3 - 189619*a^2 - 217635*a + 56419],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()
 
\(\mathfrak{D}\) = \((4747561509943,4747561509943 a^{5} - 4747561509943 a^{4} - 23737807549715 a^{3} + 18990246039772 a^{2} + 23737807549715 a - 9495123019886,966462779485 a^{5} - 966462779484 a^{4} - 4832313897426 a^{3} + 3865851117937 a^{2} + 4832313897427 a + 812730634258,2923202918893 a^{5} - 2923202918893 a^{4} - 14616014594464 a^{3} + 11692811675571 a^{2} + 14616014594462 a - 4423509729161,4346099027516 a^{5} - 4346099027516 a^{4} - 21730495137580 a^{3} + 17384396110064 a^{2} + 21730495137581 a - 5768995136140,3310656490286 a^{5} - 3310656490286 a^{4} - 16553282451430 a^{3} + 13242625961145 a^{2} + 16553282451430 a - 2731647282195)\) = \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right)^{15} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 22539340290692258087863249 \) = \( 49^{15} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{5071828648518898928776706}{4747561509943} a^{5} + \frac{1594371267260221471874998}{4747561509943} a^{4} + \frac{22976097914409962241364493}{4747561509943} a^{3} - \frac{1845254644248882604992608}{4747561509943} a^{2} - \frac{13259741923351079752225853}{4747561509943} a + \frac{3007050597491418428537932}{4747561509943} \)
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\((j)_{-}\)
\( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right) \) \(49\) \(1\) \(I_{15}\) Non-split multiplicative \(1\) \(1\) \(15\) \(15\)

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.