Invariants
Level: | $48$ | $\SL_2$-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: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16G1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.1503 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&22\\44&39\end{bmatrix}$, $\begin{bmatrix}5&19\\32&47\end{bmatrix}$, $\begin{bmatrix}33&29\\16&23\end{bmatrix}$, $\begin{bmatrix}45&37\\8&15\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.1.bk.2 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $64$ |
Full 48-torsion field degree: | $12288$ |
Jacobian
Conductor: | $2^{5}$ |
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.bk.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 |
---|---|---|---|---|---|---|
16.48.1-16.b.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.0-24.by.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-48.e.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-48.e.2.18 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-24.by.1.13 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.1-16.b.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
48.192.1-48.j.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.bc.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.bk.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.cb.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.cp.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.cs.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.de.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.dj.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.288.9-48.fa.2.3 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
48.384.9-48.bbb.2.2 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
96.192.5-96.z.2.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.bd.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.bp.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.cb.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.cn.1.7 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.cz.1.3 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.dd.1.3 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.dh.1.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.192.1-240.hx.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.ib.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.in.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.ir.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.jn.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.jv.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.kt.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.lb.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.480.17-240.cu.2.2 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |