Invariants
| Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{3}\cdot10^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.36.1.12 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&28\\20&31\end{bmatrix}$, $\begin{bmatrix}17&21\\34&29\end{bmatrix}$, $\begin{bmatrix}21&13\\34&5\end{bmatrix}$, $\begin{bmatrix}21&23\\20&19\end{bmatrix}$, $\begin{bmatrix}27&17\\14&5\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $20480$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} + 367x + 2863 $ |
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot5}\cdot\frac{7020x^{2}y^{10}+1728014700000x^{2}y^{8}z^{2}-72932960400000000x^{2}y^{6}z^{4}+588734606340000000000x^{2}y^{4}z^{6}-1408381450820000000000000x^{2}y^{2}z^{8}+631654486810000000000000000x^{2}z^{10}+16743780xy^{10}z+56747925000000xy^{8}z^{3}-478654961100000000xy^{6}z^{5}-3739779227640000000000xy^{4}z^{7}+32333713769420000000000000xy^{2}z^{9}-52189232853660000000000000000xz^{11}+y^{12}+14278242480y^{10}z^{2}-723025539300000y^{8}z^{4}+18256298291900000000y^{6}z^{6}-135044289981140000000000y^{4}z^{8}+380790385064120000000000000y^{2}z^{10}-365758121704310000000000000000z^{12}}{x^{2}y^{10}-6880000x^{2}y^{8}z^{2}-27648000000x^{2}y^{6}z^{4}-3958272000000000x^{2}y^{4}z^{6}-53063680000000000000x^{2}y^{2}z^{8}+33021952000000000000000x^{2}z^{10}-406xy^{10}z+368480000xy^{8}z^{3}+10879488000000xy^{6}z^{5}+142403072000000000xy^{4}z^{7}-407203840000000000000xy^{2}z^{9}-2999017472000000000000000xz^{11}+71209y^{10}z^{2}-11240320000y^{8}z^{4}-147487232000000y^{6}z^{6}+941125632000000000y^{4}z^{8}+6672343040000000000000y^{2}z^{10}-22611197952000000000000000z^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(10)$ | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.6.0.b.1 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.12.1.d.1 | $40$ | $3$ | $3$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 40.72.1.bf.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.72.1.bf.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.72.1.bg.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.72.1.bg.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.72.1.bi.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.72.1.bi.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.72.1.bj.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.72.1.bj.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.72.3.bn.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.72.3.bp.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.72.3.bt.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.72.3.bv.1 | $40$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
| 40.72.3.db.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.72.3.db.2 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.72.3.dc.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.72.3.dc.2 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.72.3.de.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.72.3.de.2 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.72.3.df.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.72.3.df.2 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.72.3.dw.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.72.3.dx.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.72.3.ec.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.72.3.ed.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.180.7.cd.1 | $40$ | $5$ | $5$ | $7$ | $3$ | $1^{6}$ |
| 120.72.1.gt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gt.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gu.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gw.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gx.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.cyx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cyz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.czd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.czf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ecz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ecz.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eda.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eda.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edc.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edd.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eft.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.108.7.j.1 | $120$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 120.144.7.hmm.1 | $120$ | $4$ | $4$ | $7$ | $?$ | not computed |
| 200.180.7.f.1 | $200$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 280.72.1.bj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bj.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bk.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bk.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bm.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bn.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.3.dn.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.do.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.dq.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.dr.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.el.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.el.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.em.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.em.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.eo.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.eo.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ep.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ep.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fj.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fk.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fm.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fn.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.19.x.1 | $280$ | $8$ | $8$ | $19$ | $?$ | not computed |