Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
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: | 48.96.1.2110 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&28\\4&27\end{bmatrix}$, $\begin{bmatrix}15&26\\40&3\end{bmatrix}$, $\begin{bmatrix}27&17\\40&9\end{bmatrix}$, $\begin{bmatrix}33&2\\8&9\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.1.bj.1 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: | $2^{6}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 12 x^{2} - 3 x y - z^{2} $ |
$=$ | $24 x y + 6 y^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 6 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2}\cdot\frac{12386304y^{2}z^{10}-221184y^{2}z^{8}w^{2}-43628544y^{2}z^{6}w^{4}+96429312y^{2}z^{4}w^{6}-37757232y^{2}z^{2}w^{8}+1572858y^{2}w^{10}-8388608z^{12}+6291456z^{10}w^{2}+4853760z^{8}w^{4}-1533952z^{6}w^{6}-694272z^{4}w^{8}-2098560z^{2}w^{10}+131071w^{12}}{w^{2}z^{2}(3072y^{2}z^{6}-4224y^{2}z^{4}w^{2}+336y^{2}z^{2}w^{4}-6y^{2}w^{6}+4096z^{6}w^{2}-1088z^{4}w^{4}+64z^{2}w^{6}-w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.bj.1 :
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
$0$ | $=$ | $ X^{4}-6X^{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.a.1.14 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.0-24.bz.2.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-48.f.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-48.f.2.30 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-24.bz.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.1-16.a.1.16 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.c.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.t.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.bl.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.bs.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.cg.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.cl.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.cx.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.da.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.288.9-48.ev.2.1 | $48$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
48.384.9-48.baw.2.6 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
240.192.1-240.hs.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hw.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.ii.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.im.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.ja.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.ji.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.kg.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.ko.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.480.17-240.ct.1.4 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |