Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 40.24.1.101 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&22\\28&31\end{bmatrix}$, $\begin{bmatrix}27&6\\1&9\end{bmatrix}$, $\begin{bmatrix}37&4\\2&13\end{bmatrix}$, $\begin{bmatrix}39&28\\32&35\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 80.48.1-40.br.1.1, 80.48.1-40.br.1.2, 80.48.1-40.br.1.3, 80.48.1-40.br.1.4, 80.48.1-40.br.1.5, 80.48.1-40.br.1.6, 80.48.1-40.br.1.7, 80.48.1-40.br.1.8, 240.48.1-40.br.1.1, 240.48.1-40.br.1.2, 240.48.1-40.br.1.3, 240.48.1-40.br.1.4, 240.48.1-40.br.1.5, 240.48.1-40.br.1.6, 240.48.1-40.br.1.7, 240.48.1-40.br.1.8 |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $30720$ |
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 $ | $=$ | $ 12 x y + 4 x z - y w - 2 z w $ |
| $=$ | $36 x^{2} + 4 x w + 2 y^{2} - 2 y z - 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 3 x^{3} z - 30 x^{2} y^{2} - 14 x^{2} z^{2} + 80 x y^{2} z - 7 x z^{3} + 80 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\cdot3^3\,\frac{3360960xz^{4}w-31720xz^{2}w^{3}-64480xw^{5}-2917376y^{2}z^{4}+409080y^{2}z^{2}w^{2}-13811y^{2}w^{4}-2681344yz^{5}+574040yz^{3}w^{2}+14406yzw^{4}-581824z^{6}+1158440z^{4}w^{2}-100724z^{2}w^{4}+8200w^{6}}{21600xz^{4}w-440xz^{2}w^{3}+16120xw^{5}-1600y^{2}z^{4}-1560y^{2}z^{2}w^{2}+3149y^{2}w^{4}+1600yz^{5}-2840yz^{3}w^{2}-4614yzw^{4}+1600z^{6}+4960z^{4}w^{2}-1684z^{2}w^{4}-2050w^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.1.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.12.0.o.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.bv.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
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.k.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ca.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ec.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ef.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ev.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fb.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gu.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gw.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.cp.1 | $40$ | $5$ | $5$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.144.9.et.1 | $40$ | $6$ | $6$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.240.17.ph.1 | $40$ | $10$ | $10$ | $17$ | $3$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.pr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.pv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.vv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.xs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.xu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.hf.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.dz.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.rx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vt.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.wv.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.wz.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.dz.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |