Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse and Zureick-Brown (RZB) label: | X131 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.1.32 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&2\\4&1\end{bmatrix}$, $\begin{bmatrix}1&5\\2&7\end{bmatrix}$, $\begin{bmatrix}1&6\\6&3\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3.Q_8$ |
| 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: | $64$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 4 y^{2} + 2 z^{2} - w^{2} $ |
| $=$ | $4 x^{2} + y w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 2 y^{2} z^{2} - 4 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(2z^{2}+3w^{2})^{3}}{w^{2}(2z^{2}-w^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.l.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.0.z.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1.bf.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bg.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bq.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.br.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.3.be.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.48.3.bf.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.48.3.bf.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.48.3.bg.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.48.1.em.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.en.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fi.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fj.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.cl.1 | $24$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 24.96.5.bl.1 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 40.48.1.du.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.dv.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ek.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.el.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.bf.1 | $40$ | $5$ | $5$ | $9$ | $4$ | $1^{6}\cdot2$ |
| 40.144.9.cd.1 | $40$ | $6$ | $6$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.240.17.hr.1 | $40$ | $10$ | $10$ | $17$ | $9$ | $1^{12}\cdot2^{2}$ |
| 48.48.3.be.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.48.3.bf.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.48.3.bf.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.48.3.bg.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 56.48.1.du.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.dv.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ek.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.el.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.bl.1 | $56$ | $8$ | $8$ | $13$ | $4$ | $1^{8}\cdot2^{2}$ |
| 56.504.37.cl.1 | $56$ | $21$ | $21$ | $37$ | $21$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.672.49.cl.1 | $56$ | $28$ | $28$ | $49$ | $25$ | $1^{12}\cdot2^{16}\cdot4$ |
| 80.48.3.bm.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.bn.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.bn.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.bo.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.48.1.du.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.dv.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.ek.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.el.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.288.21.bl.1 | $88$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 104.48.1.du.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.dv.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.ek.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.el.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.3.be.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.bf.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.bf.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.bg.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.1.mc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.md.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ni.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.nj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.du.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.dv.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.ek.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.el.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.du.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.dv.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.ek.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.el.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.mc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.md.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ni.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.nj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.3.be.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.48.3.bf.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.48.3.bf.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.48.3.bg.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.48.1.du.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.dv.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.ek.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.el.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.3.bm.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.48.3.bn.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.48.3.bn.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.48.3.bo.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.48.1.du.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.dv.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.ek.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.el.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.3.bm.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.bn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.bn.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.bo.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.48.1.du.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.dv.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.ek.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.el.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.mc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.md.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ni.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.nj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.3.bm.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.48.3.bn.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.48.3.bn.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.48.3.bo.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.48.1.le.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.lf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.mk.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ml.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.du.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.dv.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.ek.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.el.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.3.be.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.48.3.bf.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.48.3.bf.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.48.3.bg.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.1.mc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.md.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.ni.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.nj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.du.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.dv.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.ek.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.el.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |