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
magma: E := ChangeRing(EllipticCurve([a^4 - 3*a^2 + 1, -a^5 + 2*a^4 + 4*a^3 - 6*a^2 - 3*a + 1, a^5 - a^4 - 4*a^3 + 3*a^2 + 2*a - 1, a^5 - 8*a^4 + 51*a^3 - 29*a^2 - 63*a + 5, 120*a^5 - 261*a^4 - 176*a^3 + 398*a^2 + 152*a - 73]),K);
sage: E = EllipticCurve(K, [a^4 - 3*a^2 + 1, -a^5 + 2*a^4 + 4*a^3 - 6*a^2 - 3*a + 1, a^5 - a^4 - 4*a^3 + 3*a^2 + 2*a - 1, a^5 - 8*a^4 + 51*a^3 - 29*a^2 - 63*a + 5, 120*a^5 - 261*a^4 - 176*a^3 + 398*a^2 + 152*a - 73])
gp (2.8): E = ellinit([a^4 - 3*a^2 + 1, -a^5 + 2*a^4 + 4*a^3 - 6*a^2 - 3*a + 1, a^5 - a^4 - 4*a^3 + 3*a^2 + 2*a - 1, a^5 - 8*a^4 + 51*a^3 - 29*a^2 - 63*a + 5, 120*a^5 - 261*a^4 - 176*a^3 + 398*a^2 + 152*a - 73],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((3,-a^{2} + 2)\) | = | \( \left(a^{2} - 2\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 9 \) | = | \( 9 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((387420489,387420489 a^{5} - 387420489 a^{4} - 1937102445 a^{3} + 1549681956 a^{2} + 1937102445 a - 774840978,301229517 a^{5} - 301229516 a^{4} - 1506147586 a^{3} + 1204918065 a^{2} + 1506147587 a - 505290178,8876348 a^{5} - 8876348 a^{4} - 44381739 a^{3} + 35505391 a^{2} + 44381737 a + 95825244,122454286 a^{5} - 122454286 a^{4} - 612271430 a^{3} + 489817144 a^{2} + 612271431 a - 236032225,133432170 a^{5} - 133432170 a^{4} - 667160850 a^{3} + 533728681 a^{2} + 667160850 a + 43241524)\) | = | \( \left(a^{2} - 2\right)^{18} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 150094635296999121 \) | = | \( 9^{18} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{19534593968251450184}{387420489} a^{5} - \frac{25732253879599264147}{387420489} a^{4} - \frac{95706620059011517651}{387420489} a^{3} + \frac{90934603429118494397}{387420489} a^{2} + \frac{101153218100845110134}{387420489} a - \frac{1059714138202501808}{14348907} \) | ||
| 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: | $\left(-\frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{7}{4} a^{2} + 3 a + \frac{13}{4} : \frac{13}{8} a^{5} + \frac{1}{8} a^{4} - \frac{15}{2} a^{3} + \frac{5}{2} a^{2} + \frac{47}{8} a - \frac{23}{8} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} - 2\right) \) | \(9\) | \(2\) | \(I_{18}\) | Non-split multiplicative | \(1\) | \(1\) | \(18\) | \(18\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
9.1-a
consists of curves linked by isogenies of
degrees dividing 6.