Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot8$ | Cusp orbits | $1^{2}$ | ||
| 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: | 8A1 |
| Rouse and Zureick-Brown (RZB) label: | X52 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.12.1.2 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}5&3\\2&1\end{bmatrix}$, $\begin{bmatrix}5&7\\6&1\end{bmatrix}$, $\begin{bmatrix}7&1\\6&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3.D_8$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 8-isogeny field degree: | $4$ |
| Cyclic 8-torsion field degree: | $16$ |
| Full 8-torsion field degree: | $128$ |
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 |
|---|
| $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{28x^{2}z^{2}-80xy^{2}z+64y^{4}+z^{4}}{z^{2}x^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.6.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 8.24.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.24.1.e.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.24.1.j.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.24.1.k.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.q.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.r.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.u.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.v.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.3.a.1 | $24$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 24.48.3.a.1 | $24$ | $4$ | $4$ | $3$ | $0$ | $1^{2}$ |
| 40.24.1.q.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.r.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.u.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.v.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.60.5.a.1 | $40$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.72.5.a.1 | $40$ | $6$ | $6$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.120.9.dc.1 | $40$ | $10$ | $10$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 56.24.1.q.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.1.r.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.1.u.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.1.v.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.7.a.1 | $56$ | $8$ | $8$ | $7$ | $1$ | $1^{6}$ |
| 56.252.19.a.1 | $56$ | $21$ | $21$ | $19$ | $1$ | $1^{2}\cdot2^{6}\cdot4$ |
| 56.336.25.a.1 | $56$ | $28$ | $28$ | $25$ | $2$ | $1^{8}\cdot2^{6}\cdot4$ |
| 88.24.1.q.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.24.1.r.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.24.1.u.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.24.1.v.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.144.11.a.1 | $88$ | $12$ | $12$ | $11$ | $?$ | not computed |
| 104.24.1.q.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.24.1.r.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.24.1.u.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.24.1.v.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.168.13.a.1 | $104$ | $14$ | $14$ | $13$ | $?$ | not computed |
| 120.24.1.q.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.r.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.u.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.v.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.24.1.q.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.24.1.r.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.24.1.u.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.24.1.v.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.216.17.a.1 | $136$ | $18$ | $18$ | $17$ | $?$ | not computed |
| 152.24.1.q.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.24.1.r.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.24.1.u.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.24.1.v.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.240.19.a.1 | $152$ | $20$ | $20$ | $19$ | $?$ | not computed |
| 168.24.1.q.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.r.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.u.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.v.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.24.1.q.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.24.1.r.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.24.1.u.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.24.1.v.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.288.23.a.1 | $184$ | $24$ | $24$ | $23$ | $?$ | not computed |
| 232.24.1.q.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.24.1.r.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.24.1.u.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.24.1.v.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.24.1.q.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.24.1.r.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.24.1.u.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.24.1.v.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.q.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.r.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.u.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.v.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.q.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.r.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.u.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.v.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.24.1.q.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.24.1.r.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.24.1.u.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.24.1.v.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.q.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.r.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.u.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.v.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.24.1.q.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.24.1.r.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.24.1.u.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.24.1.v.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |