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: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.412 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&12\\14&9\end{bmatrix}$, $\begin{bmatrix}9&6\\11&23\end{bmatrix}$, $\begin{bmatrix}39&14\\4&33\end{bmatrix}$ |
| 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} - 3 y^{2} - 2 y z - 2 z^{2} $ |
| $=$ | $5 x^{2} + 4 y^{2} + 6 y z + 6 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 12 x^{2} y^{2} - 20 x^{2} z^{2} + 196 y^{4} - 420 y^{2} z^{2} + 225 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{233766000000yz^{11}+1167075000000yz^{9}w^{2}-4842127080000yz^{7}w^{4}+3116065820000yz^{5}w^{6}+591905564600yz^{3}w^{8}-515529514500yzw^{10}+535221000000z^{12}-1406295000000z^{10}w^{2}-1881774090000z^{8}w^{4}+4862237660000z^{6}w^{6}-2202519174100z^{4}w^{8}+87861473700z^{2}w^{10}+49147147049w^{12}}{2164500000yz^{11}-917000000yz^{9}w^{2}+959420000yz^{7}w^{4}-482601000yz^{5}w^{6}+152103350yz^{3}w^{8}+5546310yzw^{10}+4955750000z^{12}-5605100000z^{10}w^{2}+3307535000z^{8}w^{4}-1037575000z^{6}w^{6}+138717775z^{4}w^{8}+25834760z^{2}w^{10}+151263w^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.1.be.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.0.cx.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.dd.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.dy.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.ej.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.bf.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.bt.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.lk.1 | $40$ | $5$ | $5$ | $17$ | $7$ | $1^{14}\cdot2$ |
| 40.288.17.bdk.1 | $40$ | $6$ | $6$ | $17$ | $6$ | $1^{14}\cdot2$ |
| 40.480.33.bts.1 | $40$ | $10$ | $10$ | $33$ | $14$ | $1^{28}\cdot2^{2}$ |
| 80.96.3.oz.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.pb.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.sn.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.so.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.sp.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.sq.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.td.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.tf.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.5.nv.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.nw.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.fja.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.buk.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.96.3.btd.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.btf.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvd.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bve.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvf.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvg.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvt.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.bpl.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bpm.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |