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: | 40.48.1.337 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}13&27\\12&19\end{bmatrix}$, $\begin{bmatrix}25&28\\16&29\end{bmatrix}$, $\begin{bmatrix}39&38\\18&13\end{bmatrix}$ |
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} + 10 y^{2} - z^{2} + z w - w^{2} $ |
$=$ | $10 x^{2} - 3 z^{2} + 2 z w - 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 36 x^{4} + 12 x^{2} y^{2} + 20 x^{2} z^{2} + y^{4} + 25 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^{14}\,\frac{w^{3}z^{3}(z^{2}-zw+w^{2})^{3}}{(z-w)^{4}(z^{2}+w^{2})^{4}}$ |
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.bc.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.0.cm.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.cs.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dq.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.eg.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.bb.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.1.bk.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.ks.1 | $40$ | $5$ | $5$ | $17$ | $6$ | $1^{14}\cdot2$ |
40.288.17.bbm.1 | $40$ | $6$ | $6$ | $17$ | $5$ | $1^{14}\cdot2$ |
40.480.33.bru.1 | $40$ | $10$ | $10$ | $33$ | $13$ | $1^{28}\cdot2^{2}$ |
120.144.9.ffg.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.btc.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |