Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| 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: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.1.1145 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&0\\36&31\end{bmatrix}$, $\begin{bmatrix}23&36\\2&5\end{bmatrix}$, $\begin{bmatrix}39&24\\12&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.h.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $3840$ |
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 $ | $=$ | $ 2 x^{2} - 2 z^{2} - w^{2} $ |
| $=$ | $x^{2} - 2 x w - 2 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 2 x^{2} y^{2} - 12 x^{2} z^{2} + 4 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 to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{11524964352xz^{22}w+131125133312xz^{20}w^{3}+559345844224xz^{18}w^{5}+1316776366080xz^{16}w^{7}+1993853575168xz^{14}w^{9}+2089570924544xz^{12}w^{11}+1568560736256xz^{10}w^{13}+851139791360xz^{8}w^{15}+329339305216xz^{6}w^{17}+87005395392xz^{4}w^{19}+14190810432xz^{2}w^{21}+1086679440xw^{23}-1593413632z^{24}-46383316992z^{22}w^{2}-307595692032z^{20}w^{4}-985761542144z^{18}w^{6}-1921750957824z^{16}w^{8}-2530468251648z^{14}w^{10}-2371525650176z^{12}w^{12}-1619709513216z^{10}w^{14}-808500681840z^{8}w^{16}-289767151168z^{6}w^{18}-71172324168z^{4}w^{20}-10802816688z^{2}w^{22}-768398401w^{24}}{z^{4}(2z^{2}+w^{2})^{2}(1050624xz^{14}w+26377728xz^{12}w^{3}+180662272xz^{10}w^{5}+523172992xz^{8}w^{7}+754244736xz^{6}w^{9}+569087776xz^{4}w^{11}+214828480xz^{2}w^{13}+31988856xw^{15}-82944z^{16}-6538752z^{14}w^{2}-79822144z^{12}w^{4}-357074240z^{10}w^{6}-763883952z^{8}w^{8}-871093824z^{6}w^{10}-543002732z^{4}w^{12}-174526212z^{2}w^{14}-22619537w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.96.1.h.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2X^{2}Y^{2}-12X^{2}Z^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.96.0-8.e.2.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.e.2.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.f.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.f.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.h.2.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.h.2.6 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.i.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.i.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-8.i.2.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.i.2.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.j.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.j.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.k.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.k.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.960.33-40.cz.1.6 | $40$ | $5$ | $5$ | $33$ | $7$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.kd.1.7 | $40$ | $6$ | $6$ | $33$ | $5$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.nz.1.11 | $40$ | $10$ | $10$ | $65$ | $13$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 80.384.5-16.c.2.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-16.j.2.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-16.p.2.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-16.w.2.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.bc.2.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.cy.2.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.df.2.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.ew.2.7 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.9-16.bt.1.5 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-16.bw.1.5 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.jx.1.5 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.kc.1.5 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.5-48.k.2.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.bm.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.bt.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.cy.2.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dg.2.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.hy.2.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.if.2.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ou.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.9-48.gj.1.15 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-48.go.1.15 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bkb.1.27 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bkm.1.27 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |