Invariants
Level: | $48$ | $\SL_2$-level: | $8$ | Newform level: | $2304$ | ||
Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $8^{6}$ | Cusp orbits | $2\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $2$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8B2 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.2.140 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&28\\44&7\end{bmatrix}$, $\begin{bmatrix}11&11\\20&21\end{bmatrix}$, $\begin{bmatrix}43&46\\28&15\end{bmatrix}$, $\begin{bmatrix}47&34\\18&17\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 48-isogeny field degree: | $32$ |
Cyclic 48-torsion field degree: | $512$ |
Full 48-torsion field degree: | $24576$ |
Jacobian
Conductor: | $2^{14}\cdot3^{2}$ |
Simple: | no |
Squarefree: | yes |
Decomposition: | $1^{2}$ |
Newforms: | 256.2.a.a, 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
$ 0 $ | $=$ | $ x z - y w $ |
$=$ | $3 z^{2} + w^{2} - w t$ | |
$=$ | $x w - x t + 3 y z$ | |
$=$ | $64 x^{2} + 48 y^{2} - w^{2} + 2 w t + t^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{6} + 12 x^{4} z^{2} + 3 x^{2} y^{2} z^{2} + 2 x^{2} z^{4} + 4 y^{2} z^{4} $ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ -8x^{6} - 54x^{4} - 72x^{2} - 27 $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 4y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Birational map from embedded model to Weierstrass model:
$\displaystyle X$ | $=$ | $\displaystyle -zw$ |
$\displaystyle Y$ | $=$ | $\displaystyle -12yz^{4}w-16yz^{2}w^{3}$ |
$\displaystyle Z$ | $=$ | $\displaystyle -z^{2}$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^2\,\frac{104485552128y^{8}+25153929216y^{6}t^{2}+2235009024y^{4}t^{4}+87304704y^{2}t^{6}+1943757w^{8}-1033560w^{7}t-286308w^{6}t^{2}-1189224w^{5}t^{3}+816798w^{4}t^{4}+194456w^{3}t^{5}-176196w^{2}t^{6}-964696wt^{7}+1009901t^{8}}{34828517376y^{8}+644972544y^{6}t^{2}-11446272y^{4}t^{4}+148992y^{2}t^{6}+20655w^{8}-192456w^{7}t+653076w^{6}t^{2}-901944w^{5}t^{3}+249834w^{4}t^{4}+331784w^{3}t^{5}-54732w^{2}t^{6}-8584wt^{7}+7343t^{8}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
16.24.1.f.1 | $16$ | $2$ | $2$ | $1$ | $1$ | $1$ |
24.24.1.de.1 | $24$ | $2$ | $2$ | $1$ | $1$ | $1$ |
48.24.0.k.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
48.96.3.r.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
48.96.3.do.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
48.96.3.hb.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
48.96.3.he.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
48.96.3.sr.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
48.96.3.sy.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1$ |
48.96.3.tn.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
48.96.3.tq.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
48.144.10.ja.1 | $48$ | $3$ | $3$ | $10$ | $7$ | $1^{4}\cdot2^{2}$ |
48.192.11.fg.1 | $48$ | $4$ | $4$ | $11$ | $4$ | $1^{7}\cdot2$ |
240.96.3.cuz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.cvh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.cwv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.cxd.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.dor.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.doz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.dqn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.dqv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.240.18.ga.1 | $240$ | $5$ | $5$ | $18$ | $?$ | not computed |
240.288.19.wgn.1 | $240$ | $6$ | $6$ | $19$ | $?$ | not computed |