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: | 40.24.1.105 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&16\\2&3\end{bmatrix}$, $\begin{bmatrix}9&20\\25&27\end{bmatrix}$, $\begin{bmatrix}17&20\\0&29\end{bmatrix}$, $\begin{bmatrix}21&2\\25&31\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 80.48.1-40.bj.1.1, 80.48.1-40.bj.1.2, 80.48.1-40.bj.1.3, 80.48.1-40.bj.1.4, 80.48.1-40.bj.1.5, 80.48.1-40.bj.1.6, 80.48.1-40.bj.1.7, 80.48.1-40.bj.1.8, 240.48.1-40.bj.1.1, 240.48.1-40.bj.1.2, 240.48.1-40.bj.1.3, 240.48.1-40.bj.1.4, 240.48.1-40.bj.1.5, 240.48.1-40.bj.1.6, 240.48.1-40.bj.1.7, 240.48.1-40.bj.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 $ | $=$ | $ 10 x y + 2 x z + z w $ |
| $=$ | $156 x^{2} - 4 x w - 10 y^{2} + 8 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 4 x^{3} z + 10 x^{2} y^{2} - 19 x^{2} z^{2} - 20 x z^{3} - 100 y^{2} z^{2} - 5 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{5}z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6\cdot3^3}{5^4}\cdot\frac{9383937264xz^{4}w+7659247440xz^{2}w^{3}+65037500xw^{5}-2987974080y^{2}z^{4}-2022742800y^{2}z^{2}w^{2}-772753000y^{2}w^{4}+1208039040yz^{3}w^{2}+628352400yzw^{4}+1722051864z^{6}+1427296572z^{4}w^{2}+839840570z^{2}w^{4}+6265625w^{6}}{1752192xz^{4}w-1560000xz^{2}w^{3}-104060xw^{5}+4623840y^{2}z^{4}-1404000y^{2}z^{2}w^{2}-100250y^{2}w^{4}+486720yz^{3}w^{2}+63960yzw^{4}-3699072z^{6}+539136z^{4}w^{2}-203798z^{2}w^{4}-10025w^{6}}$ |
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 |
| 40.12.0.br.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.bt.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.l.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ca.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.dm.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.do.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fm.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fo.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gd.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gj.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.ch.1 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.144.9.dv.1 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
| 40.240.17.nt.1 | $40$ | $10$ | $10$ | $17$ | $5$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.pb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.pf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ql.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.vg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.vi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.fb.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.db.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.qr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.qv.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.rh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.rl.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.uj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.un.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.uz.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.db.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |