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 | $2^{2}\cdot4^{3}$ | ||
| 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: | 40.192.1.770 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&12\\36&15\end{bmatrix}$, $\begin{bmatrix}1&20\\28&3\end{bmatrix}$, $\begin{bmatrix}3&12\\6&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.1.i.1 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 $ | $=$ | $ y^{2} + y z - z^{2} + w^{2} $ |
| $=$ | $10 x^{2} + 3 y^{2} - 2 y z + 2 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 6 x^{2} y^{2} - 20 x^{2} z^{2} + 4 y^{4} - 60 y^{2} z^{2} + 225 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^2}\cdot\frac{11320312500000yz^{23}-49809375000000yz^{21}w^{2}+95090625000000yz^{19}w^{4}-103241250000000yz^{17}w^{6}+70220242968750yz^{15}w^{8}-31082920312500yz^{13}w^{10}+9016934062500yz^{11}w^{12}-1686155625000yz^{9}w^{14}+195043944375yz^{7}w^{16}-12918701250yz^{5}w^{18}+421687350yz^{3}w^{20}-4688460yzw^{22}-6996337890625z^{24}+35846484375000z^{22}w^{2}-80032148437500z^{20}w^{4}+102282343750000z^{18}w^{6}-82643753906250z^{16}w^{8}+44004021093750z^{14}w^{10}-15616353593750z^{12}w^{12}+3655861687500z^{10}w^{14}-547098911250z^{8}w^{16}+49340526875z^{6}w^{18}-2409394275z^{4}w^{20}+50923410z^{2}w^{22}-226981w^{24}}{w^{8}(15421875yz^{15}-43181250yz^{13}w^{2}+48116250yz^{11}w^{4}-27142500yz^{9}w^{6}+8140650yz^{7}w^{8}-1241100yz^{5}w^{10}+82068yz^{3}w^{12}-1512yzw^{14}-9531250z^{16}+33584375z^{14}w^{2}-47669375z^{12}w^{4}+34982750z^{10}w^{6}-14135075z^{8}w^{8}+3083450z^{6}w^{10}-327862z^{4}w^{12}+13068z^{2}w^{14}-81w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.i.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+6X^{2}Y^{2}+4Y^{4}-20X^{2}Z^{2}-60Y^{2}Z^{2}+225Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.0-40.i.2.7 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.i.2.13 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.k.2.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.k.2.14 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.u.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.u.1.13 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.w.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.w.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-8.i.2.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.m.2.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.m.2.9 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.p.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.p.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.cd.1.2 | $40$ | $5$ | $5$ | $33$ | $5$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.hl.2.3 | $40$ | $6$ | $6$ | $33$ | $3$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.jn.1.1 | $40$ | $10$ | $10$ | $65$ | $7$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 80.384.9-80.gp.2.5 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.gq.2.5 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.gu.2.5 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.gv.2.5 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ut.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.uu.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.vg.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.vh.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |