Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| 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: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.1.136 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&0\\0&7\end{bmatrix}$, $\begin{bmatrix}5&0\\4&3\end{bmatrix}$, $\begin{bmatrix}5&2\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.i.2 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $16$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 11x - 14 $ |
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 |
|---|
| $(-2:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{24x^{2}y^{14}+424786x^{2}y^{12}z^{2}+1011536664x^{2}y^{10}z^{4}+588578534409x^{2}y^{8}z^{6}+109936935701376x^{2}y^{6}z^{8}+8365764142695129x^{2}y^{4}z^{10}+273771076691951604x^{2}y^{2}z^{12}+3203095830928031745x^{2}z^{14}+1036xy^{14}z+6584712xy^{12}z^{3}+10157502399xy^{10}z^{5}+4099344479562xy^{8}z^{7}+605868487863256xy^{6}z^{9}+39304781176502856xy^{4}z^{11}+1145262787644096537xy^{2}z^{13}+12262818962286313470xz^{15}+y^{16}+20112y^{14}z^{2}+86989668y^{12}z^{4}+78616007544y^{10}z^{6}+20141247406412y^{8}z^{8}+2007992773878816y^{6}z^{10}+89258362532785194y^{4}z^{12}+1745005629946200144y^{2}z^{14}+11713254600860499961z^{16}}{zy^{4}(287x^{2}y^{8}z+66800x^{2}y^{6}z^{3}+2293821x^{2}y^{4}z^{5}+4x^{2}y^{2}z^{7}+x^{2}z^{9}+xy^{10}+2390xy^{8}z^{2}+325308xy^{6}z^{4}+8781712xy^{4}z^{6}-7xy^{2}z^{8}-2xz^{10}+24y^{10}z+13046y^{8}z^{3}+776976y^{6}z^{5}+8388142y^{4}z^{7}-32y^{2}z^{9}-7z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.d.2.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.d.2.10 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.1-8.c.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1-8.c.1.10 | $8$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 8.192.1-8.b.2.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.c.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.g.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.h.2.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.5-16.m.2.7 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.192.5-16.n.2.5 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.192.5-16.p.2.7 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 16.192.5-16.q.2.5 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.i.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.j.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.t.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.u.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.df.2.10 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.bs.1.9 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 40.192.1-40.i.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.j.2.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.t.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.u.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.bb.2.8 | $40$ | $5$ | $5$ | $17$ | $3$ | $1^{6}\cdot2^{5}$ |
| 40.576.17-40.cc.2.1 | $40$ | $6$ | $6$ | $17$ | $1$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.960.33-40.fh.2.2 | $40$ | $10$ | $10$ | $33$ | $5$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 48.192.5-48.bj.2.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.192.5-48.bk.1.13 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.192.5-48.bp.2.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.5-48.bq.1.13 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 56.192.1-56.i.1.5 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.j.2.3 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.t.2.2 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.u.1.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.768.25-56.bs.1.1 | $56$ | $8$ | $8$ | $25$ | $1$ | $1^{8}\cdot2^{4}\cdot4^{2}$ |
| 56.2016.73-56.df.2.5 | $56$ | $21$ | $21$ | $73$ | $9$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2688.97-56.df.2.1 | $56$ | $28$ | $28$ | $97$ | $10$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 80.192.5-80.bj.2.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.bk.1.9 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.bp.2.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.bq.1.9 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.192.1-88.i.1.3 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.j.2.3 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.t.2.3 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.u.2.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.i.1.5 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.j.2.7 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.t.2.3 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.u.1.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.5-112.bj.2.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.bk.1.13 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.bp.2.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.bq.1.13 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.1-120.be.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.bf.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.cd.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ce.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.i.1.3 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.j.2.7 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.t.2.3 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.u.1.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.i.1.3 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.j.2.3 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.t.2.5 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.u.1.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.be.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.bf.2.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.cd.2.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ce.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.5-176.bj.2.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5-176.bk.1.13 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5-176.bp.2.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5-176.bq.1.13 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.192.1-184.i.1.5 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.j.2.3 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.t.2.3 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.u.2.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.5-208.bj.2.2 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5-208.bk.1.9 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5-208.bp.2.2 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5-208.bq.1.9 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.192.1-232.i.1.3 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.j.2.7 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.t.2.3 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.u.1.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.5-240.eb.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ec.1.29 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.eo.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ep.1.29 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.192.1-248.i.1.5 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.j.2.3 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.t.2.2 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.u.2.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.be.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.bf.2.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.cd.2.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ce.2.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.5-272.bj.2.4 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5-272.bk.2.11 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5-272.bp.2.4 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5-272.bq.2.10 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.1-280.be.2.10 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.bf.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.cd.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ce.2.6 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.i.1.3 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.j.2.7 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.t.2.5 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.u.1.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.5-304.bj.2.2 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5-304.bk.1.13 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5-304.bp.2.2 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5-304.bq.1.13 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.1-312.be.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.bf.2.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.cd.2.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ce.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.i.1.3 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.j.2.7 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.t.2.5 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.u.1.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |