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: | $4$ (all of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.1.39 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}5&2\\0&7\end{bmatrix}$, $\begin{bmatrix}5&2\\4&5\end{bmatrix}$, $\begin{bmatrix}7&0\\4&3\end{bmatrix}$, $\begin{bmatrix}7&2\\0&3\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.1.d.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} - 4x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(2:0:1)$, $(0:0:1)$, $(-2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{48x^{2}y^{4}z^{2}+4xy^{6}z+768xy^{2}z^{5}+y^{8}+4096z^{8}}{z^{2}y^{4}x^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-4.b.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.24.0-4.b.1.11 | $8$ | $2$ | $2$ | $0$ | $0$ | 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.g.1.9 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.l.1.6 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.m.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.m.2.6 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.n.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.p.1.7 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.u.1.13 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.w.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.x.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.x.2.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.y.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ba.1.13 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.144.5-24.d.1.20 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.d.1.32 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 40.96.1-40.u.1.14 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.w.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.x.1.14 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.x.2.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.y.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.ba.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.240.9-40.d.1.4 | $40$ | $5$ | $5$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.288.9-40.d.1.22 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
| 40.480.17-40.bv.1.4 | $40$ | $10$ | $10$ | $17$ | $4$ | $1^{12}\cdot2^{2}$ |
| 56.96.1-56.u.1.12 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.w.1.10 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.x.1.6 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.x.2.12 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.y.1.12 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.ba.1.11 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.384.13-56.d.1.20 | $56$ | $8$ | $8$ | $13$ | $3$ | $1^{8}\cdot2^{2}$ |
| 56.1008.37-56.d.1.8 | $56$ | $21$ | $21$ | $37$ | $9$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.1344.49-56.d.1.19 | $56$ | $28$ | $28$ | $49$ | $12$ | $1^{12}\cdot2^{16}\cdot4$ |
| 88.96.1-88.u.1.14 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-88.w.1.10 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-88.x.1.10 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-88.x.2.12 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-88.y.1.12 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-88.ba.1.7 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.u.1.12 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.w.1.6 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.x.1.14 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.x.2.6 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.y.1.5 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-104.ba.1.8 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.u.1.31 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.w.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.x.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.x.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.y.1.26 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ba.1.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.u.1.14 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.w.1.10 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.x.1.6 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.x.2.14 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.y.1.9 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.96.1-136.ba.1.12 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.u.1.14 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.w.1.10 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.x.1.10 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.x.2.12 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.y.1.12 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.96.1-152.ba.1.11 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.u.1.30 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.w.1.20 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.x.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.x.2.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.y.1.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ba.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.u.1.14 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.w.1.10 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.x.1.10 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.x.2.12 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.y.1.12 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.96.1-184.ba.1.7 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.u.1.14 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.w.1.10 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.x.1.6 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.x.2.14 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.y.1.9 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.96.1-232.ba.1.12 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.u.1.14 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.w.1.10 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.x.1.10 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.x.2.12 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.y.1.12 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-248.ba.1.7 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.u.1.28 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.w.1.20 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.x.1.12 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.x.2.20 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.y.1.20 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.ba.1.16 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.u.1.31 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.w.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.x.1.12 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.x.2.6 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.y.1.28 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.ba.1.32 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.u.1.14 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.w.1.10 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.x.1.6 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.x.2.14 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.y.1.9 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.96.1-296.ba.1.12 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.u.1.27 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.w.1.20 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.x.1.12 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.x.2.14 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.y.1.18 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ba.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.u.1.12 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.w.1.10 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.x.1.6 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.x.2.14 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.y.1.9 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.96.1-328.ba.1.12 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |