Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $8^{12}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.3.96 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}15&34\\8&1\end{bmatrix}$, $\begin{bmatrix}25&34\\24&15\end{bmatrix}$, $\begin{bmatrix}37&12\\20&21\end{bmatrix}$, $\begin{bmatrix}39&24\\16&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.3.be.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: | $2^{18}\cdot5^{6}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 1600.2.a.n, 1600.2.d.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x z t - x w t - y w t - y t^{2} $ |
| $=$ | $2 x z w - x w^{2} - y w^{2} - y w t$ | |
| $=$ | $2 x z^{2} - x z w - y z w - y z t$ | |
| $=$ | $2 x y z - x y w - y^{2} w - y^{2} t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} z + 5 x^{3} y^{2} - 5 x^{3} z^{2} + 10 x^{2} y^{2} z - 6 x^{2} z^{3} - 10 x y^{2} z^{2} + \cdots - 20 y^{2} z^{3} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -5x^{7} + 35x^{5} - 35x^{3} + 5x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:-1/2:-1:1)$, $(0:1:0:0:0)$, $(1:0:0:0:0)$, $(-1:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{78125x^{14}+93750x^{12}t^{2}+68750x^{10}t^{4}-15000x^{8}t^{6}+142750x^{6}t^{8}-280800x^{4}t^{10}+624540x^{2}t^{12}+156250xy^{13}+718750xy^{11}t^{2}+175000xy^{9}t^{4}+775000xy^{7}t^{6}+849250xy^{5}t^{8}+2752550xy^{3}t^{10}+6047680xyt^{12}+312500y^{12}t^{2}-50000y^{10}t^{4}+555000y^{8}t^{6}+1048000y^{6}t^{8}+2574700y^{4}t^{10}+6906400y^{2}t^{12}-640zw^{13}+128zw^{12}t-4352zw^{11}t^{2}+256zw^{10}t^{3}-43904zw^{9}t^{4}+4480zw^{8}t^{5}-232960zw^{7}t^{6}+382152zw^{6}t^{7}-1223424zw^{5}t^{8}+2461408zw^{4}t^{9}-3658176zw^{3}t^{10}+5776040zw^{2}t^{11}-4838272zwt^{12}+640zt^{13}+224w^{14}-64w^{13}t+1568w^{12}t^{2}-384w^{11}t^{3}+15456w^{10}t^{4}-4032w^{9}t^{5}+83104w^{8}t^{6}-153428w^{7}t^{7}+430884w^{6}t^{8}-1026920w^{5}t^{9}+1319744w^{4}t^{10}-2783260w^{3}t^{11}+1862948w^{2}t^{12}-1631160wt^{13}+224t^{14}}{t^{4}(125x^{6}t^{4}-250x^{4}t^{6}+590x^{2}t^{8}+250xy^{5}t^{4}-50xy^{3}t^{6}+2960xyt^{8}+500y^{4}t^{6}+1360y^{2}t^{8}-40zw^{9}+8zw^{8}t-272zw^{7}t^{2}+16zw^{6}t^{3}-904zw^{5}t^{4}-88zw^{4}t^{5}-2048zw^{3}t^{6}-184zw^{2}t^{7}-2368zwt^{8}+14w^{10}-4w^{9}t+98w^{8}t^{2}-24w^{7}t^{3}+322w^{6}t^{4}-68w^{5}t^{5}+686w^{4}t^{6}-236w^{3}t^{7}+732w^{2}t^{8}-508wt^{9})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.3.be.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{3}Y^{2}-X^{4}Z+10X^{2}Y^{2}Z-5X^{3}Z^{2}-10XY^{2}Z^{2}-6X^{2}Z^{3}-20Y^{2}Z^{3}-2XZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.96.3.be.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x-y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x^{3}t+2x^{2}yt-2xy^{2}t-4y^{3}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.c.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.c.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-40.n.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.96.1-40.n.1.10 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.96.2-40.a.1.3 | $40$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 40.96.2-40.a.1.15 | $40$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.384.5-40.z.1.3 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.384.5-40.z.1.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.384.5-40.z.2.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.384.5-40.z.2.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.384.5-40.bb.3.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.384.5-40.bb.3.8 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.384.5-40.bb.4.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.384.5-40.bb.4.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.960.35-40.bo.1.8 | $40$ | $5$ | $5$ | $35$ | $5$ | $1^{14}\cdot2^{5}\cdot4^{2}$ |
| 40.1152.37-40.es.1.18 | $40$ | $6$ | $6$ | $37$ | $4$ | $1^{14}\cdot2^{2}\cdot4^{4}$ |
| 40.1920.69-40.gu.1.18 | $40$ | $10$ | $10$ | $69$ | $8$ | $1^{28}\cdot2^{7}\cdot4^{6}$ |
| 80.384.7-80.d.1.4 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.d.1.16 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.g.1.12 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.g.1.15 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.z.1.5 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.z.1.14 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.ba.1.12 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.ba.1.15 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.cj.1.12 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.cj.1.15 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.ck.1.10 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.ck.1.16 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.dd.1.8 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.dd.1.14 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.dg.1.10 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.384.7-80.dg.1.16 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.5-120.hl.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hl.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hl.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hl.2.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hn.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hn.1.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hn.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hn.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.u.1.15 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.u.1.26 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.x.1.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.x.1.27 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ck.1.15 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ck.1.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.cl.1.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.cl.1.24 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.fq.1.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.fq.1.24 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.fr.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.fr.1.31 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.he.1.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.he.1.27 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.hh.1.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.hh.1.27 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.384.5-280.gz.1.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.gz.1.11 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.gz.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.gz.2.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.ha.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.ha.1.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.ha.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.ha.2.10 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |