Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.1.84 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}3&0\\0&5\end{bmatrix}$, $\begin{bmatrix}7&0\\4&3\end{bmatrix}$, $\begin{bmatrix}7&0\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^4$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.g.1 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: | $2^{6}$ |
| 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 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{70x^{2}y^{12}z^{2}-4551x^{2}y^{8}z^{6}+27645x^{2}y^{4}z^{10}-4095x^{2}z^{14}-8xy^{14}z+1599xy^{10}z^{5}-25604xy^{6}z^{9}+20481xy^{2}z^{13}+y^{16}-308y^{12}z^{4}+10292y^{8}z^{8}-16386y^{4}z^{12}+z^{16}}{z^{2}y^{8}(x^{2}y^{4}-15x^{2}z^{4}+17xy^{2}z^{3}-6y^{4}z^{2}+z^{6})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-4.b.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-4.b.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.h.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.h.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.1-8.d.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1-8.d.1.9 | $8$ | $2$ | $2$ | $1$ | $0$ | 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.a.1.6 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.a.2.4 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.f.1.6 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.f.2.4 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.3-8.e.1.4 | $8$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 8.192.3-8.f.1.6 | $8$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 8.192.3-8.f.2.8 | $8$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 8.192.3-8.g.2.6 | $8$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.1-24.e.1.8 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.e.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.p.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.p.2.8 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.3-24.o.2.10 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.192.3-24.p.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.p.2.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.q.1.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.288.9-24.cr.1.34 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 24.384.9-24.bh.2.42 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
| 40.192.1-40.e.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.e.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.p.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.p.2.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.3-40.t.2.9 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.u.1.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.192.3-40.u.2.9 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.192.3-40.v.1.13 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.480.17-40.v.1.21 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.576.17-40.bs.1.25 | $40$ | $6$ | $6$ | $17$ | $2$ | $1^{14}\cdot2$ |
| 40.960.33-40.et.1.29 | $40$ | $10$ | $10$ | $33$ | $10$ | $1^{28}\cdot2^{2}$ |
| 56.192.1-56.e.1.4 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.e.2.8 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.p.1.4 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.p.2.8 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.3-56.l.2.11 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.192.3-56.m.1.14 | $56$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 56.192.3-56.m.2.9 | $56$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 56.192.3-56.n.1.13 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.768.25-56.bh.1.30 | $56$ | $8$ | $8$ | $25$ | $6$ | $1^{20}\cdot2^{2}$ |
| 56.2016.73-56.cr.1.9 | $56$ | $21$ | $21$ | $73$ | $23$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.2688.97-56.cr.2.24 | $56$ | $28$ | $28$ | $97$ | $29$ | $1^{36}\cdot2^{28}\cdot4$ |
| 88.192.1-88.e.1.8 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.e.2.4 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.p.1.6 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.p.2.8 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.3-88.l.2.9 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.192.3-88.m.1.14 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.192.3-88.m.2.9 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.192.3-88.n.1.13 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.192.1-104.e.1.4 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.e.2.8 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.p.1.4 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.p.2.8 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.3-104.t.2.9 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.192.3-104.u.1.14 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.192.3-104.u.2.9 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.192.3-104.v.1.13 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.1-120.u.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.u.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.bt.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.bt.2.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.bp.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.bq.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.bq.2.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.br.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.192.1-136.e.1.8 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.e.2.4 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.p.1.6 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.p.2.8 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.3-136.t.2.9 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.192.3-136.u.1.14 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.192.3-136.u.2.9 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.192.3-136.v.1.13 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.192.1-152.e.1.8 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.e.2.4 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.p.1.6 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.p.2.8 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.3-152.l.2.11 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.192.3-152.m.1.14 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.192.3-152.m.2.9 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.192.3-152.n.1.13 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.1-168.u.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.u.2.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.bt.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.bt.2.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.bh.2.27 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.bi.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.bi.2.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.bj.1.27 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.192.1-184.e.1.8 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.e.2.4 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.p.1.6 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.p.2.8 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.3-184.l.2.9 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.192.3-184.m.1.14 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.192.3-184.m.2.9 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.192.3-184.n.1.13 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.192.1-232.e.1.8 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.e.2.4 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.p.1.6 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.p.2.8 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.3-232.t.2.9 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.192.3-232.u.1.14 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.192.3-232.u.2.9 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.192.3-232.v.1.13 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.192.1-248.e.1.8 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.e.2.4 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.p.1.6 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.p.2.8 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.3-248.l.2.11 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.192.3-248.m.1.14 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.192.3-248.m.2.9 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.192.3-248.n.1.13 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.u.1.12 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.u.2.16 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.bt.1.16 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.bt.2.12 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.3-264.bh.2.19 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.bi.1.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.bi.2.22 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.bj.1.27 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.192.1-280.u.1.16 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.u.2.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.bt.1.12 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.bt.2.16 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.3-280.bp.1.24 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.192.3-280.bq.1.32 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.192.3-280.bq.2.28 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.192.3-280.br.1.32 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.192.1-296.e.1.8 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.e.2.4 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.p.1.6 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.p.2.8 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.3-296.t.2.9 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.192.3-296.u.1.14 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.192.3-296.u.2.9 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.192.3-296.v.1.13 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.1-312.u.1.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.u.2.12 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.bt.1.12 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.bt.2.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.bp.2.28 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.bq.1.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.bq.2.22 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.br.1.28 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.192.1-328.e.1.8 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.e.2.4 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.p.1.6 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.p.2.8 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.3-328.t.2.11 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.192.3-328.u.1.14 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.192.3-328.u.2.9 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.192.3-328.v.1.13 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |