Invariants
Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16G1 |
Level structure
$\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}43&158\\26&39\end{bmatrix}$, $\begin{bmatrix}136&191\\53&150\end{bmatrix}$, $\begin{bmatrix}175&108\\232&91\end{bmatrix}$, $\begin{bmatrix}192&47\\167&40\end{bmatrix}$, $\begin{bmatrix}206&153\\143&220\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.1.bm.2 for the level structure with $-I$) |
Cyclic 240-isogeny field degree: | $48$ |
Cyclic 240-torsion field degree: | $3072$ |
Full 240-torsion field degree: | $5898240$ |
Jacobian
Conductor: | $?$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 24 x y - 6 y^{2} + w^{2} $ |
$=$ | $24 x^{2} + 6 x y - z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 3 x^{2} y^{2} - 9 x^{2} z^{2} + 18 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{387072y^{2}z^{10}+13824y^{2}z^{8}w^{2}-5453568y^{2}z^{6}w^{4}-24107328y^{2}z^{4}w^{6}-18878616y^{2}z^{2}w^{8}-1572858y^{2}w^{10}-131072z^{12}-196608z^{10}w^{2}+303360z^{8}w^{4}+191744z^{6}w^{6}-173568z^{4}w^{8}+1049280z^{2}w^{10}+131071w^{12}}{w^{2}z^{2}(384y^{2}z^{6}+1056y^{2}z^{4}w^{2}+168y^{2}z^{2}w^{4}+6y^{2}w^{6}-512z^{6}w^{2}-272z^{4}w^{4}-32z^{2}w^{6}-w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.bm.2 :
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
$0$ | $=$ | $ X^{4}-3X^{2}Y^{2}-9X^{2}Z^{2}+18Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
80.48.1-16.b.1.15 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.0-24.by.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
240.48.0-48.f.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
240.48.0-48.f.1.21 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
240.48.0-24.by.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
240.48.1-16.b.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | 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 |
---|---|---|---|---|---|---|
240.192.1-48.j.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-48.bd.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-48.bo.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-48.cb.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-48.cn.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-48.cu.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-48.dg.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-48.dh.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hz.2.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.id.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.ip.2.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.it.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.jp.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.jx.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.kv.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.ld.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.288.9-48.fk.1.11 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
240.384.9-48.bbd.2.23 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
240.480.17-240.da.1.13 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |