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: | 40.48.1.414 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&30\\4&7\end{bmatrix}$, $\begin{bmatrix}23&4\\10&3\end{bmatrix}$, $\begin{bmatrix}35&14\\7&5\end{bmatrix}$ |
| 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 $ | $=$ | $ 3 y^{2} - 2 y w - 2 z^{2} + 2 w^{2} $ |
| $=$ | $5 x^{2} - y^{2} - y w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 100 x^{4} - 60 x^{2} y^{2} - 30 x^{2} z^{2} + 9 y^{4} + 4 y^{2} z^{2} + 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 5x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 5w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^8}\cdot\frac{2857680yz^{10}w+89067600yz^{8}w^{3}-500688000yz^{6}w^{5}+1017360000yz^{4}w^{7}-924000000yz^{2}w^{9}+308000000yw^{11}+83349z^{12}+21296520z^{10}w^{2}-78426900z^{8}w^{4}+79560000z^{6}w^{6}-15930000z^{4}w^{8}+21200000z^{2}w^{10}-23000000w^{12}}{z^{8}(60yz^{2}w-100yw^{3}+12z^{4}-10z^{2}w^{2}+25w^{4})}$ |
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.be.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.0.cu.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.db.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.dw.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.ef.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.be.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.bv.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.lg.1 | $40$ | $5$ | $5$ | $17$ | $9$ | $1^{14}\cdot2$ |
| 40.288.17.bdg.1 | $40$ | $6$ | $6$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.480.33.bto.1 | $40$ | $10$ | $10$ | $33$ | $18$ | $1^{28}\cdot2^{2}$ |
| 80.96.3.ov.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.ox.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.sj.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.sk.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.sl.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.sm.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.sz.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.tb.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.5.nt.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.nu.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.fiw.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.bug.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.96.3.bsz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.btb.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.buz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bva.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvb.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvc.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvp.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bvr.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.bpj.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bpk.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |