Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $320$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $2^{4}$ | ||
| 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: | 20J3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.1252 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&19\\2&35\end{bmatrix}$, $\begin{bmatrix}11&26\\2&25\end{bmatrix}$, $\begin{bmatrix}21&15\\28&13\end{bmatrix}$, $\begin{bmatrix}23&38\\22&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.3.t.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{16}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 80.2.a.b, 320.2.a.d, 320.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x u + y t $ |
| $=$ | $z t - 3 w t - w u$ | |
| $=$ | $x z - 3 x w - y w$ | |
| $=$ | $3 x^{2} + 2 x y - y^{2} - z w + w^{2} + t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{8} - 80 x^{6} y^{2} + 8 x^{6} z^{2} + 676 x^{4} y^{4} - 52 x^{4} y^{2} z^{2} + x^{4} z^{4} + \cdots - 2 y^{2} z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{8} + 16x^{7} + 12x^{6} + 16x^{5} - 38x^{4} - 16x^{3} + 12x^{2} - 16x - 1 $ |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{44049458934xw^{9}-215816284008xw^{7}u^{2}+176101818048xw^{5}u^{4}-43658844064xw^{3}u^{6}-821033898919xwt^{8}-484033720326xwt^{7}u+2392856356657xwt^{6}u^{2}+2094862715192xwt^{5}u^{3}-2160013988782xwt^{4}u^{4}-2466380615924xwt^{3}u^{5}-68938423222xwt^{2}u^{6}+987350991000xwtu^{7}+460887857365xwu^{8}+4078653605yzu^{8}+22024729467yw^{9}-56299900176yw^{7}u^{2}+20079402408yw^{5}u^{4}+12077551520yw^{3}u^{6}+61179804075ywu^{8}}{w(1631461442xw^{8}-405451956xw^{6}u^{2}-130255736xw^{4}u^{4}+112644288xw^{2}u^{6}-35608483872xt^{8}+24442275312xt^{7}u+16979470560xt^{6}u^{2}-6350373744xt^{5}u^{3}+125303776xt^{4}u^{4}+588968864xt^{3}u^{5}-161118336xt^{2}u^{6}-16042560xtu^{7}+815730721yw^{8}-318569394yw^{6}u^{2}+45961240yw^{4}u^{4}+8021280yw^{2}u^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.t.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{8}-80X^{6}Y^{2}+676X^{4}Y^{4}+8X^{6}Z^{2}-52X^{4}Y^{2}Z^{2}+416X^{2}Y^{4}Z^{2}+X^{4}Z^{4}-16X^{2}Y^{2}Z^{4}+64Y^{4}Z^{4}-2Y^{2}Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.t.1 :
| $\displaystyle X$ | $=$ | $\displaystyle \frac{26}{29}ywt^{2}+\frac{12}{29}ywtu+\frac{2}{29}ywu^{2}-\frac{52}{29}yt^{3}-\frac{24}{29}yt^{2}u-\frac{4}{29}ytu^{2}+\frac{13}{29}wt^{2}u+\frac{6}{29}wtu^{2}+\frac{1}{29}wu^{3}+\frac{3}{29}t^{3}u-\frac{2}{29}t^{2}u^{2}-\frac{1}{29}tu^{3}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{8959808}{707281}ywt^{11}u^{3}-\frac{15886464}{707281}ywt^{10}u^{4}-\frac{11411840}{707281}ywt^{9}u^{5}-\frac{4135040}{707281}ywt^{8}u^{6}-\frac{652800}{707281}ywt^{7}u^{7}+\frac{43648}{707281}ywt^{6}u^{8}+\frac{36224}{707281}ywt^{5}u^{9}+\frac{5760}{707281}ywt^{4}u^{10}+\frac{320}{707281}ywt^{3}u^{11}+\frac{7037888}{707281}yt^{12}u^{3}+\frac{14553472}{707281}yt^{11}u^{4}+\frac{430720}{24389}yt^{10}u^{5}+\frac{5587328}{707281}yt^{9}u^{6}+\frac{43520}{24389}yt^{8}u^{7}+\frac{67712}{707281}yt^{7}u^{8}-\frac{31872}{707281}yt^{6}u^{9}-\frac{7040}{707281}yt^{5}u^{10}-\frac{448}{707281}yt^{4}u^{11}-\frac{2887456}{707281}wt^{11}u^{4}-\frac{6562624}{707281}wt^{10}u^{5}-\frac{6355392}{707281}wt^{9}u^{6}-\frac{3390272}{707281}wt^{8}u^{7}-\frac{1068800}{707281}wt^{7}u^{8}-\frac{196544}{707281}wt^{6}u^{9}-\frac{18496}{707281}wt^{5}u^{10}-\frac{448}{707281}wt^{4}u^{11}+\frac{32}{707281}wt^{3}u^{12}-\frac{2182224}{707281}t^{12}u^{4}-\frac{1714144}{707281}t^{11}u^{5}+\frac{1065632}{707281}t^{10}u^{6}+\frac{1750048}{707281}t^{9}u^{7}+\frac{854400}{707281}t^{8}u^{8}+\frac{203104}{707281}t^{7}u^{9}+\frac{22624}{707281}t^{6}u^{10}+\frac{608}{707281}t^{5}u^{11}-\frac{48}{707281}t^{4}u^{12}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t^{3}u-\frac{10}{29}t^{2}u^{2}-\frac{1}{29}tu^{3}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-8.h.1.1 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 40.72.1-20.b.1.11 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.288.5-40.ch.1.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.ch.2.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.ci.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.ci.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.co.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.co.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.cp.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.cp.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.720.19-40.ex.1.6 | $40$ | $5$ | $5$ | $19$ | $6$ | $1^{16}$ |
| 80.288.7-80.k.1.6 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.k.1.12 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.l.1.7 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.l.1.14 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.m.1.2 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.m.1.16 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.n.1.5 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.n.1.16 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.o.1.6 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.o.1.12 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.o.2.8 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.o.2.10 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.p.1.6 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.p.1.12 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.p.2.8 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.p.2.10 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.5-120.ni.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ni.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.nj.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.nj.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.np.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.np.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.nq.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.nq.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.15-120.cv.1.23 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 240.288.7-240.k.1.14 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.k.1.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.l.1.15 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.l.1.22 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.m.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.m.1.30 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.n.1.10 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.n.1.31 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.o.1.7 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.o.1.28 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.o.2.15 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.o.2.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.p.1.7 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.p.1.28 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.p.2.15 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.p.2.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.5-280.gp.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.gp.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.gq.1.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.gq.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.gw.1.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.gw.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.gx.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.gx.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |