Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.1.22 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&3\\4&3\end{bmatrix}$, $\begin{bmatrix}1&7\\0&3\end{bmatrix}$, $\begin{bmatrix}7&7\\4&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_4\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.1.n.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $8$ |
| Full 8-torsion field degree: | $32$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{723x^{2}y^{4}z^{2}-4095x^{2}z^{6}-46xy^{6}z+8193xy^{2}z^{5}+y^{8}-4140y^{4}z^{4}+z^{8}}{zy^{4}(x^{2}z+xy^{2}+z^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.24.0-4.d.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.24.0-4.d.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.z.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.bc.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.bl.1.2 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.bm.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.3-16.bc.1.4 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.96.3-16.bc.2.4 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.96.3-16.bd.1.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3-16.bd.2.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.1-24.ea.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ee.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ew.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.fa.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.144.5-24.bx.1.8 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.bd.1.15 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 40.96.1-40.di.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.dm.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.dy.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.ec.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.240.9-40.z.1.3 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.288.9-40.bt.1.16 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
| 40.480.17-40.hd.1.14 | $40$ | $10$ | $10$ | $17$ | $6$ | $1^{12}\cdot2^{2}$ |
| 48.96.3-48.bc.1.4 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.96.3-48.bc.2.4 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.96.3-48.bd.1.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3-48.bd.2.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 56.96.1-56.di.1.4 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.dm.1.4 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.dy.1.4 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.ec.1.3 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.384.13-56.bd.1.20 | $56$ | $8$ | $8$ | $13$ | $4$ | $1^{8}\cdot2^{2}$ |
| 56.1008.37-56.bx.1.14 | $56$ | $21$ | $21$ | $37$ | $12$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.1344.49-56.bx.1.15 | $56$ | $28$ | $28$ | $49$ | $16$ | $1^{12}\cdot2^{16}\cdot4$ |
| 80.96.3-80.bk.1.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-80.bk.2.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-80.bl.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-80.bl.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.96.1-88.di.1.4 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-88.dm.1.4 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-88.dy.1.4 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-88.ec.1.3 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.di.1.3 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.dm.1.3 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.dy.1.3 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.ec.1.3 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.3-112.bc.1.6 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3-112.bc.2.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3-112.bd.1.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3-112.bd.2.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.1-120.la.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.li.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.mg.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.mo.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.di.1.4 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.dm.1.4 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.dy.1.3 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.ec.1.4 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.di.1.4 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.dm.1.4 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.dy.1.4 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.ec.1.4 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.la.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.li.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.mg.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.mo.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.3-176.bc.1.6 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3-176.bc.2.4 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3-176.bd.1.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3-176.bd.2.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.96.1-184.di.1.3 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.dm.1.4 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.dy.1.4 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.ec.1.4 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.3-208.bk.1.6 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3-208.bk.2.4 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3-208.bl.1.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3-208.bl.2.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.96.1-232.di.1.4 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.dm.1.4 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.dy.1.3 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.ec.1.4 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.3-240.bk.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-240.bk.2.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-240.bl.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-240.bl.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.96.1-248.di.1.3 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.dm.1.4 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.dy.1.4 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.ec.1.4 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.la.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.li.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.mg.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.mo.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.3-272.bk.1.4 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3-272.bk.2.7 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3-272.bl.1.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3-272.bl.2.3 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.96.1-280.kc.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.kk.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.li.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.lq.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.di.1.4 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.dm.1.4 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.dy.1.3 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.ec.1.4 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.3-304.bc.1.6 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3-304.bc.2.4 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3-304.bd.1.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3-304.bd.2.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.1-312.la.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.li.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.mg.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.mo.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.di.1.3 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.dm.1.2 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.dy.1.2 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.ec.1.3 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |