Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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: | 8C1 |
| Rouse and Zureick-Brown (RZB) label: | X139 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.1.5 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&3\\0&7\end{bmatrix}$, $\begin{bmatrix}1&6\\2&3\end{bmatrix}$, $\begin{bmatrix}5&6\\2&3\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3.D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $8$ |
| Full 8-torsion field degree: | $64$ |
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 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{12x^{2}y^{6}-2433x^{2}y^{4}z^{2}+608256x^{2}y^{2}z^{4}-131493887x^{2}z^{6}-82xy^{6}z+46344xy^{4}z^{3}-1658879xy^{2}z^{5}-503414786xz^{7}+y^{8}-1632y^{6}z^{2}+323440y^{4}z^{4}-28311556y^{2}z^{6}-480854023z^{8}}{z(97x^{2}y^{4}z+12032x^{2}y^{2}z^{3}+304384x^{2}z^{5}+xy^{6}+638xy^{4}z^{2}+55296xy^{2}z^{4}+1165312xz^{6}+12y^{6}z+2681y^{4}z^{3}+114688y^{2}z^{5}+1113088z^{7})}$ |
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.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.0.r.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.1.d.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.b.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.x.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bb.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bf.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gc.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gs.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gw.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.eo.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.ci.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 40.48.1.fe.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fi.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fu.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fy.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.cc.1 | $40$ | $5$ | $5$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.144.9.dq.1 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
| 40.240.17.no.1 | $40$ | $10$ | $10$ | $17$ | $5$ | $1^{12}\cdot2^{2}$ |
| 56.48.1.fe.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fi.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fu.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fy.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.ci.1 | $56$ | $8$ | $8$ | $13$ | $3$ | $1^{12}$ |
| 56.504.37.eo.1 | $56$ | $21$ | $21$ | $37$ | $13$ | $1^{8}\cdot2^{12}\cdot4$ |
| 56.672.49.eo.1 | $56$ | $28$ | $28$ | $49$ | $16$ | $1^{20}\cdot2^{12}\cdot4$ |
| 88.48.1.fe.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fi.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fu.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fy.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.288.21.ci.1 | $88$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 104.48.1.fe.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fi.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fu.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fy.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ui.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.um.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.uy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.vc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fe.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fi.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fu.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fy.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fe.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fi.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fu.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fy.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ug.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.uk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.uw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.va.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fe.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fi.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fu.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fy.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fe.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fi.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fu.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fy.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fe.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fi.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fu.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fy.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ug.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.uk.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.uw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.va.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.tk.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.to.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ua.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ue.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fe.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fi.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fu.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fy.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.ui.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.um.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.uy.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.vc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fe.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fi.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fu.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fy.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |