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: | 40.48.1.401 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}13&38\\15&19\end{bmatrix}$, $\begin{bmatrix}23&2\\16&21\end{bmatrix}$, $\begin{bmatrix}31&30\\22&21\end{bmatrix}$ |
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 $ | $=$ | $ y^{2} + y w + z^{2} - w^{2} $ |
$=$ | $10 x^{2} + 3 y^{2} - 2 y w + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 225 x^{4} - 60 x^{2} y^{2} - 20 x^{2} z^{2} + 4 y^{4} + 6 y^{2} z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 5x$ |
$\displaystyle Z$ | $=$ | $\displaystyle 5w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{5760yz^{10}w-142200yz^{8}w^{3}+1055250yz^{6}w^{5}-3279375yz^{4}w^{7}+4500000yz^{2}w^{9}-2250000yw^{11}+512z^{12}-32160z^{10}w^{2}+387300z^{8}w^{4}-1796875z^{6}w^{6}+3838125z^{4}w^{8}-3787500z^{2}w^{10}+1390625w^{12}}{z^{8}(30yz^{2}w-75yw^{3}+9z^{4}-55z^{2}w^{2}+50w^{4})}$ |
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.bf.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.0.cl.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.cr.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dn.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.ef.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.ba.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.1.bl.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.kr.1 | $40$ | $5$ | $5$ | $17$ | $9$ | $1^{14}\cdot2$ |
40.288.17.bbl.1 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{14}\cdot2$ |
40.480.33.brt.1 | $40$ | $10$ | $10$ | $33$ | $16$ | $1^{28}\cdot2^{2}$ |
80.96.3.og.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.oi.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.rs.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.ru.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.144.9.fff.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.btb.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
240.96.3.bsk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.bsm.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.bui.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.buk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |