Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
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: | 40.48.1.327 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&9\\18&19\end{bmatrix}$, $\begin{bmatrix}13&26\\10&23\end{bmatrix}$, $\begin{bmatrix}17&15\\20&39\end{bmatrix}$, $\begin{bmatrix}21&5\\6&23\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 40.96.1-40.ge.1.1, 40.96.1-40.ge.1.2, 80.96.1-40.ge.1.1, 80.96.1-40.ge.1.2, 80.96.1-40.ge.1.3, 80.96.1-40.ge.1.4, 120.96.1-40.ge.1.1, 120.96.1-40.ge.1.2, 240.96.1-40.ge.1.1, 240.96.1-40.ge.1.2, 240.96.1-40.ge.1.3, 240.96.1-40.ge.1.4, 280.96.1-40.ge.1.1, 280.96.1-40.ge.1.2 |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $384$ |
Full 40-torsion field degree: | $15360$ |
Jacobian
Conductor: | $2^{6}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 y^{2} - 2 y z + 2 z^{2} + w^{2} $ |
$=$ | $20 x^{2} - y^{2} - y z + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 4 x^{2} y^{2} + 15 x^{2} z^{2} + 9 y^{4} + 30 y^{2} z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^8}{3^4}\cdot\frac{4812500yz^{11}+7218750yz^{9}w^{2}+2835000yz^{7}w^{4}-47250yz^{5}w^{6}-1668600yz^{3}w^{8}-466560yzw^{10}-359375z^{12}-165625z^{10}w^{2}+4066875z^{8}w^{4}+4685625z^{6}w^{6}+2070900z^{4}w^{8}+272160z^{2}w^{10}-124416w^{12}}{w^{4}(5000yz^{7}+4500yz^{5}w^{2}-900yz^{3}w^{4}-540yzw^{6}-18125z^{8}-21250z^{6}w^{2}-7275z^{4}w^{4}-810z^{2}w^{6}+27w^{8})}$ |
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.ba.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
20.24.0.g.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.cr.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dq.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.ei.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.y.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.1.bg.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.ki.1 | $40$ | $5$ | $5$ | $17$ | $8$ | $1^{14}\cdot2$ |
40.288.17.bbc.1 | $40$ | $6$ | $6$ | $17$ | $4$ | $1^{14}\cdot2$ |
40.480.33.brk.1 | $40$ | $10$ | $10$ | $33$ | $13$ | $1^{28}\cdot2^{2}$ |
80.96.3.nx.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.nz.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.rj.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.rl.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.144.9.few.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.bss.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
240.96.3.bsb.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.bsd.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.btz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.bub.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |