Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8C1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.7 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&2\\22&17\end{bmatrix}$, $\begin{bmatrix}9&26\\6&15\end{bmatrix}$, $\begin{bmatrix}9&36\\16&5\end{bmatrix}$, $\begin{bmatrix}31&39\\22&29\end{bmatrix}$ |
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: | $30720$ |
Jacobian
Conductor: | $2^{5}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 800.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x z + y w $ |
$=$ | $8 x^{2} + 5 y^{2} + 5 z^{2} + 4 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} + 5 x^{2} y^{2} + x^{2} z^{2} + 2 y^{2} z^{2} $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{3375y^{6}+18900y^{4}w^{2}+56880y^{2}w^{4}-60625z^{6}-27000z^{4}w^{2}+14400z^{2}w^{4}-1728w^{6}}{125y^{6}-500y^{4}w^{2}+80y^{2}w^{4}+125z^{6}+200z^{4}w^{2}-64w^{6}}$ |
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 |
---|---|---|---|---|---|---|
4.12.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.12.0.bu.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.12.1.h.1 | $40$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
40.48.1.b.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.48.1.cl.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.48.1.dw.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.48.1.ei.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.48.1.hk.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.48.1.hu.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.48.1.jm.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.48.1.ka.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.120.9.ea.1 | $40$ | $5$ | $5$ | $9$ | $4$ | $1^{6}\cdot2$ |
40.144.9.iu.1 | $40$ | $6$ | $6$ | $9$ | $3$ | $1^{6}\cdot2$ |
40.240.17.uw.1 | $40$ | $10$ | $10$ | $17$ | $6$ | $1^{12}\cdot2^{2}$ |
120.48.1.bda.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bdi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bew.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bfe.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.chk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.chq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.cje.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.cjo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.72.5.bsi.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
120.96.5.qu.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.bfq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bfu.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bgw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bha.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bpm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bpq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bqs.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bqw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.192.13.jw.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |