Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $24$ | $\PSL_2$-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 and Zureick-Brown (RZB) label: | X134 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.1.10 |
Level structure
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
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 |
|---|---|---|---|---|---|---|
| 4.12.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.0.z.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.1.c.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.z.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bc.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bl.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bm.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.3.bc.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.48.3.bc.2 | $16$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 16.48.3.bd.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.48.3.bd.2 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.1.ea.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ee.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ew.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fa.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.bx.1 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.96.5.bd.1 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 40.48.1.di.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.dm.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.dy.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ec.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.z.1 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.144.9.bt.1 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
| 40.240.17.hd.1 | $40$ | $10$ | $10$ | $17$ | $6$ | $1^{12}\cdot2^{2}$ |
| 48.48.3.bc.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.48.3.bc.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.48.3.bd.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.48.3.bd.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 56.48.1.di.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.dm.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.dy.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ec.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.bd.1 | $56$ | $8$ | $8$ | $13$ | $4$ | $1^{8}\cdot2^{2}$ |
| 56.504.37.bx.1 | $56$ | $21$ | $21$ | $37$ | $12$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.672.49.bx.1 | $56$ | $28$ | $28$ | $49$ | $16$ | $1^{12}\cdot2^{16}\cdot4$ |
| 80.48.3.bk.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.bk.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.bl.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.bl.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.48.1.di.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.dm.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.dy.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.ec.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.288.21.bd.1 | $88$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 104.48.1.di.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.dm.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.dy.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.ec.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.3.bc.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.bc.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.bd.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.bd.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.1.la.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.li.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.mg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.mo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.di.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.dm.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.dy.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.ec.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.di.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.dm.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.dy.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.ec.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.la.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.li.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.mg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.mo.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.3.bc.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.48.3.bc.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.48.3.bd.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.48.3.bd.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.48.1.di.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.dm.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.dy.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.ec.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.3.bk.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.48.3.bk.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.48.3.bl.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.48.3.bl.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.48.1.di.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.dm.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.dy.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.ec.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.3.bk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.bk.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.bl.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.bl.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.48.1.di.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.dm.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.dy.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.ec.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.la.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.li.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.mg.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.mo.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.3.bk.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.48.3.bk.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.48.3.bl.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.48.3.bl.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.48.1.kc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.kk.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.li.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.lq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.di.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.dm.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.dy.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.ec.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.3.bc.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.48.3.bc.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.48.3.bd.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.48.3.bd.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.1.la.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.li.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.mg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.mo.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.di.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.dm.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.dy.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.ec.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |