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 $4$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.1.153 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\4&3\end{bmatrix}$, $\begin{bmatrix}3&6\\4&5\end{bmatrix}$, $\begin{bmatrix}7&4\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.n.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} - 44x + 112 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(4:0:1)$, $(6:8:1)$, $(6:-8:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^4}\cdot\frac{48x^{2}y^{14}-356896x^{2}y^{12}z^{2}+701893632x^{2}y^{10}z^{4}-723570779136x^{2}y^{8}z^{6}+443515503378432x^{2}y^{6}z^{8}-165078270613192704x^{2}y^{4}z^{10}+35042697816563515392x^{2}y^{2}z^{12}-3279970130870308700160x^{2}z^{14}-1264xy^{14}z+5240064xy^{12}z^{3}-8619949824xy^{10}z^{5}+7846356996096xy^{8}z^{7}-4372637482614784xy^{6}z^{9}+1496846267870871552xy^{4}z^{11}-293187273636914921472xy^{2}z^{13}+25114253234762353213440xz^{15}-y^{16}+22656y^{14}z^{2}-57368832y^{12}z^{4}+71446622208y^{10}z^{6}-51548108275712y^{8}z^{8}+22961101307117568y^{6}z^{10}-6169291573720252416y^{4}z^{12}+893442882532622204928y^{2}z^{14}-47977490845124490428416z^{16}}{z^{2}y^{4}(x^{2}y^{8}-22688x^{2}y^{6}z^{2}+40288320x^{2}y^{4}z^{4}-19413336064x^{2}y^{2}z^{6}+2710594125824x^{2}z^{8}-48xy^{8}z+347536xy^{6}z^{3}-428539648xy^{4}z^{5}+169198223360xy^{2}z^{7}-20754624151552xz^{9}+1224y^{8}z^{2}-3698176y^{6}z^{4}+2583689472y^{4}z^{6}-598711730176y^{2}z^{8}+39648990593024z^{10})}$ |
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.e.1.11 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.2.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.2.9 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.1-8.d.1.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1-8.d.1.6 | $8$ | $2$ | $2$ | $1$ | $0$ | 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.f.1.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.f.2.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.j.1.4 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.j.2.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bg.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bg.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bo.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bo.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.ei.1.10 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.cm.1.22 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.192.1-40.bg.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bg.2.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bo.1.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bo.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.bo.1.2 | $40$ | $5$ | $5$ | $17$ | $2$ | $1^{6}\cdot2^{5}$ |
| 40.576.17-40.cx.1.15 | $40$ | $6$ | $6$ | $17$ | $0$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.960.33-40.gk.1.7 | $40$ | $10$ | $10$ | $33$ | $4$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 56.192.1-56.bg.1.6 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.bg.2.3 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.bo.1.4 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.bo.2.5 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.768.25-56.cm.1.15 | $56$ | $8$ | $8$ | $25$ | $3$ | $1^{8}\cdot2^{4}\cdot4^{2}$ |
| 56.2016.73-56.ei.1.4 | $56$ | $21$ | $21$ | $73$ | $9$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2688.97-56.ei.1.8 | $56$ | $28$ | $28$ | $97$ | $12$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 88.192.1-88.bg.1.6 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.bg.2.5 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.bo.1.7 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.bo.2.5 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.bg.1.6 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.bg.2.3 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.bo.1.7 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.bo.2.3 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eg.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eg.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ew.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ew.2.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.bg.1.6 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.bg.2.5 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.bo.1.7 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.bo.2.5 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.bg.1.6 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.bg.2.5 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.bo.1.7 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.bo.2.5 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.eg.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.eg.2.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ew.1.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ew.2.11 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.bg.1.6 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.bg.2.5 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.bo.1.7 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.bo.2.5 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.bg.1.6 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.bg.2.5 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.bo.1.7 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.bo.2.5 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.bg.1.6 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.bg.2.5 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.bo.1.7 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.bo.2.5 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.eg.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.eg.2.11 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ew.1.7 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ew.2.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.eg.1.13 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.eg.2.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ew.1.13 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ew.2.15 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.bg.1.6 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.bg.2.5 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.bo.1.7 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.bo.2.3 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.eg.1.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.eg.2.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ew.1.13 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ew.2.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.bg.1.6 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.bg.2.3 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.bo.1.7 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.bo.2.3 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |