Invariants
| Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.1.34 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}21&10\\30&11\end{bmatrix}$, $\begin{bmatrix}21&30\\32&39\end{bmatrix}$, $\begin{bmatrix}25&16\\22&29\end{bmatrix}$, $\begin{bmatrix}39&15\\6&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.1.ba.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.w |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 133x - 363 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(-3:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^5\cdot5^5}\cdot\frac{240x^{2}y^{22}+70960000x^{2}y^{20}z^{2}+530000000000x^{2}y^{18}z^{4}-23773680000000000x^{2}y^{16}z^{6}+38741600000000000000x^{2}y^{14}z^{8}+1048749280000000000000000x^{2}y^{12}z^{10}-1954973280000000000000000000x^{2}y^{10}z^{12}-11646808800000000000000000000000x^{2}y^{8}z^{14}-7973196880000000000000000000000000x^{2}y^{6}z^{16}+1637238000000000000000000000000000000x^{2}y^{4}z^{18}-100262640000000000000000000000000000000x^{2}y^{2}z^{20}+1996240000000000000000000000000000000000x^{2}z^{22}+25440xy^{22}z+2302560000xy^{20}z^{3}-9836000000000xy^{18}z^{5}-453946080000000000xy^{16}z^{7}+2593521600000000000000xy^{14}z^{9}+12594783680000000000000000xy^{12}z^{11}-48443897280000000000000000000xy^{10}z^{13}-165366820800000000000000000000000xy^{8}z^{15}-92500053280000000000000000000000000xy^{6}z^{17}+19663212000000000000000000000000000000xy^{4}z^{19}-1215711840000000000000000000000000000000xy^{2}z^{21}+24315040000000000000000000000000000000000xz^{23}+y^{24}+1606160y^{22}z^{2}+47391040000y^{20}z^{4}-800218000000000y^{18}z^{6}-3059060120000000000y^{16}z^{8}+60027082400000000000000y^{14}z^{10}-10912028480000000000000000y^{12}z^{12}-755874412320000000000000000000y^{10}z^{14}-901749728200000000000000000000000y^{8}z^{16}-103487807920000000000000000000000000y^{6}z^{18}+38036928000000000000000000000000000000y^{4}z^{20}-2621383760000000000000000000000000000000y^{2}z^{22}+54979960000000000000000000000000000000000z^{24}}{z^{3}y^{2}(y^{2}+1000z^{2})^{5}(13000x^{2}y^{6}z+421400000x^{2}y^{4}z^{3}+1703000000000x^{2}y^{2}z^{5}+1525000000000000x^{2}z^{7}+xy^{8}+578000xy^{6}z^{2}+8598400000xy^{4}z^{4}+24958000000000xy^{2}z^{6}+18575000000000000xz^{8}+173y^{8}z+17167000y^{6}z^{3}+111152600000y^{4}z^{5}+153797000000000y^{2}z^{7}+42000000000000000z^{9})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.72.0-10.a.2.4 | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.72.0-10.a.2.9 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-40.cj.1.8 | $40$ | $3$ | $3$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.288.5-40.ek.1.11 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.es.1.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.fm.1.11 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.fu.1.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.hw.1.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.ic.1.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.iy.1.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.je.1.11 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.720.13-40.cl.1.1 | $40$ | $5$ | $5$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
| 120.288.5-120.clo.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.cme.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.cns.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.coi.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ecq.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.edc.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eeu.1.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.efg.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-120.su.2.30 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 280.288.5-280.bbw.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bca.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bcy.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bdc.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bkm.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bkq.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.blo.1.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bls.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |