Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| 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: | 8F1 |
| Rouse and Zureick-Brown (RZB) label: | X262 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.1.89 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}5&0\\2&3\end{bmatrix}$, $\begin{bmatrix}7&1\\4&1\end{bmatrix}$, $\begin{bmatrix}7&5\\4&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3:C_4$ |
| 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: | $32$ |
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 $ | $=$ | $ x^{2} - y^{2} - z^{2} $ |
| $=$ | $2 x^{2} + y^{2} - 2 y z + z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 4 x^{2} y^{2} + 36 y^{4} - 12 y^{2} z^{2} + 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot3^3\,\frac{14720yz^{11}-48192yz^{9}w^{2}-544320yz^{7}w^{4}+59616yz^{5}w^{6}+911736yz^{3}w^{8}+218700yzw^{10}+2112z^{12}+150656z^{10}w^{2}+32976z^{8}w^{4}-921600z^{6}w^{6}-43092z^{4}w^{8}+398520z^{2}w^{10}+30375w^{12}}{3680yz^{11}-7296yz^{9}w^{2}+9504yz^{7}w^{4}-6264yz^{5}w^{6}+2754yz^{3}w^{8}-1458yzw^{10}+528z^{12}+1376z^{10}w^{2}-6120z^{8}w^{4}+5904z^{6}w^{6}-3915z^{4}w^{8}+1458z^{2}w^{10}+243w^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.0.v.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.24.0.z.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.24.0.bq.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.24.0.bt.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.24.1.p.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.24.1.bd.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.24.1.be.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 |
|---|---|---|---|---|---|---|
| 8.96.3.p.1 | $8$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.1.t.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.3.cj.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.dk.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3.dl.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.dm.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3.dy.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 16.96.5.cm.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.96.5.co.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 16.96.5.cq.1 | $16$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.96.3.ex.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.144.9.bkh.1 | $24$ | $3$ | $3$ | $9$ | $4$ | $1^{8}$ |
| 24.192.9.mf.1 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
| 40.96.3.bv.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.240.17.ix.1 | $40$ | $5$ | $5$ | $17$ | $9$ | $1^{14}\cdot2$ |
| 40.288.17.xd.1 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{14}\cdot2$ |
| 40.480.33.bmd.1 | $40$ | $10$ | $10$ | $33$ | $19$ | $1^{28}\cdot2^{2}$ |
| 48.96.1.ce.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.3.ii.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.jg.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.jh.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.ji.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.kg.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.5.hf.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 48.96.5.hg.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.96.5.hi.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.96.3.bd.1 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.384.25.mf.1 | $56$ | $8$ | $8$ | $25$ | $7$ | $1^{20}\cdot2^{2}$ |
| 56.1008.73.bkh.1 | $56$ | $21$ | $21$ | $73$ | $44$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.1344.97.bjn.1 | $56$ | $28$ | $28$ | $97$ | $51$ | $1^{36}\cdot2^{28}\cdot4$ |
| 80.96.1.cd.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.3.kc.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.la.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.lb.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.lc.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.mi.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.5.hh.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.hi.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.hk.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.96.3.bd.1 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.96.3.bv.1 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.1.cd.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.96.3.ia.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.iy.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.iz.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.ja.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.jy.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.5.gz.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.ha.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.hc.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.96.3.lh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.96.3.bv.1 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.96.3.bd.1 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.jb.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.1.cd.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.96.3.ia.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.iy.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.iz.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.ja.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.jy.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.5.gz.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.ha.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.hc.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.96.3.bd.1 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.1.cd.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.96.3.kc.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.la.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.lb.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.lc.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.mi.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.5.hh.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.hi.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.hk.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.96.3.bv.1 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.1.gi.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.3.bco.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bfi.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bfj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bfk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bie.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.vb.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.vc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.ve.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.96.3.bd.1 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.jb.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.1.cd.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.96.3.ju.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.la.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.lb.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.lc.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.mi.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.5.hh.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.hi.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.hk.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.96.3.et.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.96.3.bv.1 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.1.cd.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.96.3.ia.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.iy.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.iz.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.ja.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.jy.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.5.gz.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.ha.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.hc.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.96.3.lh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.96.3.bv.1 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |