Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.438 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&32\\37&7\end{bmatrix}$, $\begin{bmatrix}9&26\\8&31\end{bmatrix}$, $\begin{bmatrix}31&18\\22&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x^{2} + 2 y^{2} + 2 y z - 2 z^{2} $ |
| $=$ | $5 x^{2} - 7 y^{2} - 2 y z + 2 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 324 x^{4} - 44 x^{2} y^{2} + 270 x^{2} z^{2} + y^{4} - 10 y^{2} z^{2} + 25 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 9x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{6}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4\cdot3^3}{5^4}\cdot\frac{2136706000000yz^{11}+4082949000000yz^{9}w^{2}+2741419080000yz^{7}w^{4}+736981092000yz^{5}w^{6}+67317172200yz^{3}w^{8}+4191298020yzw^{10}-1027861000000z^{12}-1635542200000z^{10}w^{2}-676786590000z^{8}w^{4}+84555900000z^{6}w^{6}+85573883700z^{4}w^{8}+6504444180z^{2}w^{10}-129730653w^{12}}{854682400yz^{11}-761925600yz^{9}w^{2}+122635296yz^{7}w^{4}+41716296yz^{5}w^{6}-7623882yz^{3}w^{8}-708588yzw^{10}-411144400z^{12}+501471680z^{10}w^{2}-163406808z^{8}w^{4}-11422296z^{6}w^{6}+10126539z^{4}w^{8}+196830z^{2}w^{10}-236196w^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.1.p.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.0.y.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.bb.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.er.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.es.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.bk.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.bn.1 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.240.17.hb.1 | $40$ | $5$ | $5$ | $17$ | $6$ | $1^{14}\cdot2$ |
| 40.288.17.qz.1 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{14}\cdot2$ |
| 40.480.33.bch.1 | $40$ | $10$ | $10$ | $33$ | $13$ | $1^{28}\cdot2^{2}$ |
| 80.96.5.gg.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.gi.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.gk.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.gm.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.cwh.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.bbx.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.96.5.rs.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.ru.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.rw.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.ry.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |