Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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: | 40.24.1.90 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&18\\21&37\end{bmatrix}$, $\begin{bmatrix}11&32\\17&1\end{bmatrix}$, $\begin{bmatrix}23&18\\8&5\end{bmatrix}$, $\begin{bmatrix}35&38\\14&21\end{bmatrix}$ |
| 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: | $30720$ |
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 $ | $=$ | $ 5 x z + 2 x w - 2 y w - z w $ |
| $=$ | $11 x^{2} - 2 x y - x z + y^{2} + y z - z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 2 x^{3} y + 10 x^{3} z - x^{2} y^{2} + 20 x^{2} z^{2} - 20 x y z^{2} + 50 x z^{3} + \cdots + 25 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{34256250xy^{5}-28687500xy^{4}w-486000000xy^{3}w^{2}-379113750xy^{2}w^{3}+1557895500xyw^{4}-292491612xw^{5}-421875y^{6}+33750000y^{5}w+70031250y^{4}w^{2}-126292500y^{3}w^{3}-508920750y^{2}w^{4}-3375000yz^{5}-97605000yz^{4}w+156238875yz^{3}w^{2}+63324450yz^{2}w^{3}-362550780yzw^{4}+342981612yw^{5}-501175000z^{6}+57510000z^{5}w-78691500z^{4}w^{2}-103135275z^{3}w^{3}-4299390z^{2}w^{4}+8917056zw^{5}-114360300w^{6}}{1268750xy^{5}-1062500xy^{4}w-1665000xy^{3}w^{2}+4108750xy^{2}w^{3}-3052250xyw^{4}+460714xw^{5}-15625y^{6}+1250000y^{5}w-582500y^{4}w^{2}+2582500y^{3}w^{3}-451875y^{2}w^{4}-125000yz^{5}-3615000yz^{4}w-3742125yz^{3}w^{2}-5459150yz^{2}w^{3}-3524965yzw^{4}-1131714yw^{5}+78125z^{6}+2130000z^{5}w+3438000z^{4}w^{2}+3167925z^{3}w^{3}+2314205z^{2}w^{4}+1398893zw^{5}-131650w^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.12.0.o.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.bu.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.j.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.cs.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.dx.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.el.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ff.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fh.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ha.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hg.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.cr.1 | $40$ | $5$ | $5$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.144.9.ev.1 | $40$ | $6$ | $6$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.240.17.pj.1 | $40$ | $10$ | $10$ | $17$ | $4$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.px.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.xy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ye.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.hh.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.eb.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.sd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.st.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vv.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vz.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.xb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.xf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.eb.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |