Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $8^{12}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.192.3.15 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\0&5\end{bmatrix}$, $\begin{bmatrix}5&0\\4&7\end{bmatrix}$, $\begin{bmatrix}5&4\\0&3\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3$ |
| Contains $-I$: | no $\quad$ (see 8.96.3.c.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $8$ |
Jacobian
| Conductor: | $2^{18}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 64.2.a.a, 64.2.b.a |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ y z + w t $ |
| $=$ | $x y + w u - u^{2}$ | |
| $=$ | $2 x z - y z + t u$ | |
| $=$ | $x w - x u - y w + 2 z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{4} + 3 x^{4} y^{2} z^{2} + 2 x^{4} z^{4} + 8 x^{2} y^{4} z^{2} + 8 x^{2} y^{2} z^{4} + \cdots + 8 y^{4} z^{4} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ w^{2} $ | $=$ | $ 2 x^{3} z - 2 x z^{3} $ |
| $0$ | $=$ | $x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{96y^{2}t^{8}u^{2}-264y^{2}t^{6}u^{4}+536y^{2}t^{4}u^{6}-2642y^{2}t^{2}u^{8}+9582y^{2}u^{10}+w^{12}-6w^{11}u+21w^{10}u^{2}-56w^{9}u^{3}+168w^{8}u^{4}-504w^{7}u^{5}+1343w^{6}u^{6}-3114w^{5}u^{7}+6843w^{4}u^{8}-13980w^{3}u^{9}+22996w^{2}u^{10}-11832wu^{11}+64t^{12}-192t^{10}u^{2}+48t^{8}u^{4}+80t^{6}u^{6}+60t^{4}u^{8}+452t^{2}u^{10}-1816u^{12}}{u^{4}(30y^{2}t^{2}u^{4}-134y^{2}u^{6}-w^{8}+6w^{7}u-21w^{6}u^{2}+56w^{5}u^{3}-122w^{4}u^{4}+228w^{3}u^{5}-347w^{2}u^{6}+178wu^{7}-4t^{4}u^{4}-4t^{2}u^{6}+23u^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.96.3.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{4}+3X^{4}Y^{2}Z^{2}+8X^{2}Y^{4}Z^{2}+4Y^{6}Z^{2}+2X^{4}Z^{4}+8X^{2}Y^{2}Z^{4}+8Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.a.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.a.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.g.2.2 | $8$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 8.96.1-8.g.2.7 | $8$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 8.96.2-8.a.1.2 | $8$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 8.96.2-8.a.1.12 | $8$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.384.5-8.a.1.3 | $8$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 8.384.5-8.b.1.3 | $8$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 16.384.9-16.y.1.5 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 16.384.9-16.y.2.5 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 16.384.9-16.z.1.5 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 16.384.9-16.z.2.5 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 24.384.5-24.i.1.7 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 24.384.5-24.j.1.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.576.19-24.y.1.18 | $24$ | $3$ | $3$ | $19$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 24.768.21-24.t.1.21 | $24$ | $4$ | $4$ | $21$ | $1$ | $1^{8}\cdot2^{3}\cdot4$ |
| 40.384.5-40.a.1.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.384.5-40.b.1.5 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.960.35-40.n.1.11 | $40$ | $5$ | $5$ | $35$ | $9$ | $1^{14}\cdot2^{5}\cdot4^{2}$ |
| 40.1152.37-40.cd.1.16 | $40$ | $6$ | $6$ | $37$ | $2$ | $1^{14}\cdot2^{2}\cdot4^{4}$ |
| 40.1920.69-40.ct.1.5 | $40$ | $10$ | $10$ | $69$ | $14$ | $1^{28}\cdot2^{7}\cdot4^{6}$ |
| 48.384.9-48.cy.1.11 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{2}\cdot2^{2}$ |
| 48.384.9-48.cy.2.11 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{2}\cdot2^{2}$ |
| 48.384.9-48.cz.1.11 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{2}\cdot2^{2}$ |
| 48.384.9-48.cz.2.11 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{2}\cdot2^{2}$ |
| 56.384.5-56.a.1.7 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.b.1.5 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.1536.53-56.t.1.16 | $56$ | $8$ | $8$ | $53$ | $6$ | $1^{20}\cdot2^{7}\cdot4^{4}$ |
| 56.4032.151-56.y.1.7 | $56$ | $21$ | $21$ | $151$ | $23$ | $1^{16}\cdot2^{26}\cdot4^{4}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 56.5376.201-56.z.1.8 | $56$ | $28$ | $28$ | $201$ | $29$ | $1^{36}\cdot2^{33}\cdot4^{8}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 80.384.9-80.fw.1.9 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.fw.2.9 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.fx.1.9 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.fx.2.9 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 88.384.5-88.a.1.7 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.384.5-88.b.1.5 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.a.1.7 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.b.1.5 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.9-112.cy.1.11 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.cy.2.11 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.cz.1.11 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.cz.2.11 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.5-120.i.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.j.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 136.384.5-136.a.1.7 | $136$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 136.384.5-136.b.1.5 | $136$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 152.384.5-152.a.1.7 | $152$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 152.384.5-152.b.1.5 | $152$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.i.1.15 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.j.1.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.9-176.cy.1.11 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.cy.2.11 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.cz.1.11 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.cz.2.11 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 184.384.5-184.a.1.7 | $184$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.384.5-184.b.1.5 | $184$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.9-208.fw.1.9 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.fw.2.9 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.fx.1.9 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.fx.2.9 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 232.384.5-232.a.1.7 | $232$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.384.5-232.b.1.5 | $232$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.9-240.qu.1.23 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.qu.2.23 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.qv.1.23 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.qv.2.23 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 248.384.5-248.a.1.7 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-248.b.1.5 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.i.1.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.j.1.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.9-272.fw.1.9 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.fw.2.9 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.fx.1.9 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.fx.2.9 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.5-280.a.1.12 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.b.1.12 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 296.384.5-296.a.1.7 | $296$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 296.384.5-296.b.1.5 | $296$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.9-304.cy.1.11 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.cy.2.11 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.cz.1.11 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.cz.2.11 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.5-312.i.1.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.j.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 328.384.5-328.a.1.7 | $328$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 328.384.5-328.b.1.5 | $328$ | $2$ | $2$ | $5$ | $?$ | not computed |