Minimal equation
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, 11, -20, 20, -20, 11, -5], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, 11, -20, 20, -20, 11, -5]), R([0, 1, 1]))
$y^2 + (x^2 + x)y = -5x^6 + 11x^5 - 20x^4 + 20x^3 - 20x^2 + 11x - 5$
Invariants
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magma: Conductor(LSeries(C)); Factorization($1);
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| \( N \) | = | \( 644 \) | = | \( 2^{2} \cdot 7 \cdot 23 \) | |
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magma: Discriminant(C); Factorization(Integers()!$1);
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| \( \Delta \) | = | \(-2576\) | = | \( -1 \cdot 2^{4} \cdot 7 \cdot 23 \) | |
Igusa-Clebsch invariants
magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
Igusa invariants
magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
G2 invariants
magma: G2Invariants(C);
| \( I_2 \) | = | \(-78072\) | = | \( -1 \cdot 2^{3} \cdot 3 \cdot 3253 \) |
| \( I_4 \) | = | \(16499460\) | = | \( 2^{2} \cdot 3 \cdot 5 \cdot 73 \cdot 3767 \) |
| \( I_6 \) | = | \(-407047873272\) | = | \( -1 \cdot 2^{3} \cdot 3 \cdot 73 \cdot 643 \cdot 361327 \) |
| \( I_{10} \) | = | \(-10551296\) | = | \( -1 \cdot 2^{16} \cdot 7 \cdot 23 \) |
| \( J_2 \) | = | \(-9759\) | = | \( -1 \cdot 3 \cdot 3253 \) |
| \( J_4 \) | = | \(3796384\) | = | \( 2^{5} \cdot 31 \cdot 43 \cdot 89 \) |
| \( J_6 \) | = | \(-1910683600\) | = | \( -1 \cdot 2^{4} \cdot 5^{2} \cdot 7 \cdot 23 \cdot 29669 \) |
| \( J_8 \) | = | \(1058457444236\) | = | \( 2^{2} \cdot 269891 \cdot 980449 \) |
| \( J_{10} \) | = | \(-2576\) | = | \( -1 \cdot 2^{4} \cdot 7 \cdot 23 \) |
| \( g_1 \) | = | \(88516980336138032799/2576\) | ||
| \( g_2 \) | = | \(220529201888022246/161\) | ||
| \( g_3 \) | = | \(70640465629725\) |
Automorphism group
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magma: AutomorphismGroup(C); IdentifyGroup($1);
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| \(\mathrm{Aut}(X)\) | \(\simeq\) | \(V_4 \) | (GAP id : [4,2]) | ||
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magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(V_4 \) | (GAP id : [4,2]) | ||
Rational points
magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
This curve is locally solvable except over $\R$.
magma: [];
There are no rational points.
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
Number of rational Weierstrass points: \(0\)
Invariants of the Jacobian:
Analytic rank: \(0\)
magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
2-Selmer rank: \(2\)
magma: HasSquareSha(Jacobian(C));
Order of Ш*: twice a square
Regulator: 1.0
Real period: 3.9284311341098506264884274618
Tamagawa numbers: 1 (p = 2), 1 (p = 7), 1 (p = 23)
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
Torsion: \(\Z/{6}\Z\)
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $G_{3,3}$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 14.a4
Elliptic curve 46.a1
Endomorphisms
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):| \(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).