↑

Properties

Label 48.192.5.bl.2
Level $48$
Index $192$
Genus $5$
Analytic rank $0$
Cusps $24$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $1152$
Index: $192$ $\PSL_2$-index:$192$
Genus: $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps: $24$ (none of which are rational) Cusp widths $4^{8}\cdot8^{12}\cdot16^{4}$ Cusp orbits $2^{4}\cdot4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16O5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.192.5.192

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}7&8\\32&15\end{bmatrix}$, $\begin{bmatrix}11&42\\28&29\end{bmatrix}$, $\begin{bmatrix}23&32\\24&23\end{bmatrix}$, $\begin{bmatrix}35&8\\4&47\end{bmatrix}$, $\begin{bmatrix}47&14\\8&45\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.384.5-48.bl.2.1, 48.384.5-48.bl.2.2, 48.384.5-48.bl.2.3, 48.384.5-48.bl.2.4, 48.384.5-48.bl.2.5, 48.384.5-48.bl.2.6, 48.384.5-48.bl.2.7, 48.384.5-48.bl.2.8, 48.384.5-48.bl.2.9, 48.384.5-48.bl.2.10, 48.384.5-48.bl.2.11, 48.384.5-48.bl.2.12, 48.384.5-48.bl.2.13, 48.384.5-48.bl.2.14, 48.384.5-48.bl.2.15, 48.384.5-48.bl.2.16, 240.384.5-48.bl.2.1, 240.384.5-48.bl.2.2, 240.384.5-48.bl.2.3, 240.384.5-48.bl.2.4, 240.384.5-48.bl.2.5, 240.384.5-48.bl.2.6, 240.384.5-48.bl.2.7, 240.384.5-48.bl.2.8, 240.384.5-48.bl.2.9, 240.384.5-48.bl.2.10, 240.384.5-48.bl.2.11, 240.384.5-48.bl.2.12, 240.384.5-48.bl.2.13, 240.384.5-48.bl.2.14, 240.384.5-48.bl.2.15, 240.384.5-48.bl.2.16
Cyclic 48-isogeny field degree: $16$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $6144$

Jacobian

Conductor: $2^{32}\cdot3^{8}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2^{2}$
Newforms: 64.2.a.a, 576.2.k.a, 1152.2.k.a

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $ $=$ $ y w + z t $
$=$ $ - 2 y t + z^{2} - w^{2}$
$=$ $12 x^{2} - y^{2} - z^{2} - w^{2} - t^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 36 x^{4} z^{4} - 12 x^{2} y^{6} - 12 x^{2} y^{4} z^{2} - 12 x^{2} y^{2} z^{4} - 12 x^{2} z^{6} + \cdots + z^{8} $
 Copy content Toggle raw display

Rational points

This modular curve has no $\Q_p$ points for $p=31$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

$j$-invariant map of degree 192 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{y^{24}-12y^{22}t^{2}+738y^{20}t^{4}-5596y^{18}t^{6}+170607y^{16}t^{8}-643608y^{14}t^{10}+12539228y^{12}t^{12}-643608y^{10}t^{14}+38902383y^{8}t^{16}+154921508y^{6}t^{18}+431358690y^{4}t^{20}+1126957044y^{2}t^{22}-49152zw^{21}t^{2}-1146880zw^{17}t^{6}-9404416zw^{13}t^{10}-67043328zw^{9}t^{14}-358301696zw^{5}t^{18}-1478918144zwt^{22}+4096w^{24}+245760w^{20}t^{4}+2465792w^{16}t^{8}+17661952w^{12}t^{12}+108195840w^{8}t^{16}+457719808w^{4}t^{20}+t^{24}}{t^{4}(y^{20}-8y^{18}t^{2}-36y^{16}t^{4}+200y^{14}t^{6}+1222y^{12}t^{8}+200y^{10}t^{10}-16420y^{8}t^{12}-65544y^{6}t^{14}-98303y^{4}t^{16}+196608y^{2}t^{18}-3072zw^{13}t^{6}-71680zw^{9}t^{10}-446464zw^{5}t^{14}-892928zwt^{18}+256w^{16}t^{4}+15360w^{12}t^{8}+142336w^{8}t^{12}+397312w^{4}t^{16})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.1.f.2 $8$ $2$ $2$ $1$ $0$ $2^{2}$
48.96.2.a.1 $48$ $2$ $2$ $2$ $0$ $1\cdot2$
48.96.2.c.1 $48$ $2$ $2$ $2$ $0$ $1\cdot2$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.384.13.j.2 $48$ $2$ $2$ $13$ $2$ $1^{2}\cdot2^{3}$
48.384.13.k.1 $48$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.384.13.bp.1 $48$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.384.13.br.1 $48$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.384.13.dv.1 $48$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.384.13.dz.2 $48$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.384.13.ft.1 $48$ $2$ $2$ $13$ $2$ $1^{2}\cdot2^{3}$
48.384.13.fu.1 $48$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{3}$
48.576.37.bfh.2 $48$ $3$ $3$ $37$ $1$ $1^{8}\cdot2^{4}\cdot8^{2}$
48.768.41.ms.1 $48$ $4$ $4$ $41$ $1$ $1^{8}\cdot2^{6}\cdot8^{2}$
240.384.13.xn.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.xo.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.zy.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.zz.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bjg.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bjh.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bnd.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bne.1 $240$ $2$ $2$ $13$ $?$ not computed