Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = -x^4 + x^3 + 3x^2 + x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + x^3z^3 + 3x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 - 2x^4 + 6x^3 + 13x^2 + 6x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(3721\) | \(=\) | \( 61^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-3721\) | \(=\) | \( - 61^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(196\) | \(=\) | \( 2^{2} \cdot 7^{2} \) |
\( I_4 \) | \(=\) | \(6649\) | \(=\) | \( 61 \cdot 109 \) |
\( I_6 \) | \(=\) | \(304573\) | \(=\) | \( 61 \cdot 4993 \) |
\( I_{10} \) | \(=\) | \(-476288\) | \(=\) | \( - 2^{7} \cdot 61^{2} \) |
\( J_2 \) | \(=\) | \(49\) | \(=\) | \( 7^{2} \) |
\( J_4 \) | \(=\) | \(-177\) | \(=\) | \( - 3 \cdot 59 \) |
\( J_6 \) | \(=\) | \(-187\) | \(=\) | \( - 11 \cdot 17 \) |
\( J_8 \) | \(=\) | \(-10123\) | \(=\) | \( - 53 \cdot 191 \) |
\( J_{10} \) | \(=\) | \(-3721\) | \(=\) | \( - 61^{2} \) |
\( g_1 \) | \(=\) | \(-282475249/3721\) | ||
\( g_2 \) | \(=\) | \(20823873/3721\) | ||
\( g_3 \) | \(=\) | \(448987/3721\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : 1 : 1)\) | \((-1 : 1 : 2)\) | \((-2 : 2 : 1)\) | \((1 : -4 : 1)\) | \((-1 : -4 : 2)\) | \((-2 : 7 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : 1 : 1)\) | \((-1 : 1 : 2)\) | \((-2 : 2 : 1)\) | \((1 : -4 : 1)\) | \((-1 : -4 : 2)\) | \((-2 : 7 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -5 : 1)\) | \((1 : 5 : 1)\) | \((-2 : -5 : 1)\) | \((-2 : 5 : 1)\) | \((-1 : -5 : 2)\) | \((-1 : 5 : 2)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.098760\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.098760\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.098760\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.098760\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.098760\) | \(\infty\) |
\((0 : 1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 + z^3\) | \(0.098760\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.007315 \) |
Real period: | \( 28.08135 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.205419 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(61\) | \(2\) | \(2\) | \(1\) | \(1 + T + 61 T^{2}\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.40.3 | no |
\(3\) | 3.480.12 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $E_6$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.6.844596301.1 with defining polynomial:
\(x^{6} - x^{5} - 25 x^{4} - 8 x^{3} + 123 x^{2} + 126 x + 27\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = \frac{6632447}{648} b^{5} - \frac{2121977}{162} b^{4} - \frac{325981945}{1296} b^{3} - \frac{3529355}{324} b^{2} + \frac{22472053}{18} b + \frac{130538657}{144}\)
\(g_6 = -\frac{2420355499}{324} b^{5} + \frac{25668283757}{2592} b^{4} + \frac{953942534383}{5184} b^{3} + \frac{1335649351}{2592} b^{2} - \frac{66892621411}{72} b - \frac{191864719165}{288}\)
Conductor norm: 1
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.844596301.1 with defining polynomial \(x^{6} - x^{5} - 25 x^{4} - 8 x^{3} + 123 x^{2} + 126 x + 27\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{61}) \) with generator \(-\frac{2}{81} a^{5} + \frac{5}{81} a^{4} + \frac{29}{81} a^{3} + \frac{13}{81} a^{2} - \frac{1}{9} a - \frac{31}{9}\) with minimal polynomial \(x^{2} - x - 15\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.3721.1 with generator \(-\frac{4}{27} a^{5} + \frac{10}{27} a^{4} + \frac{85}{27} a^{3} - \frac{82}{27} a^{2} - \frac{44}{3} a - \frac{11}{3}\) with minimal polynomial \(x^{3} - x^{2} - 20 x + 9\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple