↑

Properties

Label 24.192.5.bs.1
Level $24$
Index $192$
Genus $5$
Analytic rank $2$
Cusps $24$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
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 $4^{12}\cdot12^{12}$ Cusp orbits $2^{2}\cdot4^{5}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12E5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.192.5.1209

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&10\\18&1\end{bmatrix}$, $\begin{bmatrix}11&12\\12&17\end{bmatrix}$, $\begin{bmatrix}13&20\\12&17\end{bmatrix}$, $\begin{bmatrix}23&0\\12&23\end{bmatrix}$, $\begin{bmatrix}23&22\\6&19\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^5.D_6$
Contains $-I$: yes
Quadratic refinements: 24.384.5-24.bs.1.1, 24.384.5-24.bs.1.2, 24.384.5-24.bs.1.3, 24.384.5-24.bs.1.4, 24.384.5-24.bs.1.5, 24.384.5-24.bs.1.6, 24.384.5-24.bs.1.7, 24.384.5-24.bs.1.8, 24.384.5-24.bs.1.9, 24.384.5-24.bs.1.10, 24.384.5-24.bs.1.11, 24.384.5-24.bs.1.12, 120.384.5-24.bs.1.1, 120.384.5-24.bs.1.2, 120.384.5-24.bs.1.3, 120.384.5-24.bs.1.4, 120.384.5-24.bs.1.5, 120.384.5-24.bs.1.6, 120.384.5-24.bs.1.7, 120.384.5-24.bs.1.8, 120.384.5-24.bs.1.9, 120.384.5-24.bs.1.10, 120.384.5-24.bs.1.11, 120.384.5-24.bs.1.12, 168.384.5-24.bs.1.1, 168.384.5-24.bs.1.2, 168.384.5-24.bs.1.3, 168.384.5-24.bs.1.4, 168.384.5-24.bs.1.5, 168.384.5-24.bs.1.6, 168.384.5-24.bs.1.7, 168.384.5-24.bs.1.8, 168.384.5-24.bs.1.9, 168.384.5-24.bs.1.10, 168.384.5-24.bs.1.11, 168.384.5-24.bs.1.12, 264.384.5-24.bs.1.1, 264.384.5-24.bs.1.2, 264.384.5-24.bs.1.3, 264.384.5-24.bs.1.4, 264.384.5-24.bs.1.5, 264.384.5-24.bs.1.6, 264.384.5-24.bs.1.7, 264.384.5-24.bs.1.8, 264.384.5-24.bs.1.9, 264.384.5-24.bs.1.10, 264.384.5-24.bs.1.11, 264.384.5-24.bs.1.12, 312.384.5-24.bs.1.1, 312.384.5-24.bs.1.2, 312.384.5-24.bs.1.3, 312.384.5-24.bs.1.4, 312.384.5-24.bs.1.5, 312.384.5-24.bs.1.6, 312.384.5-24.bs.1.7, 312.384.5-24.bs.1.8, 312.384.5-24.bs.1.9, 312.384.5-24.bs.1.10, 312.384.5-24.bs.1.11, 312.384.5-24.bs.1.12
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{24}\cdot3^{7}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2$
Newforms: 48.2.a.a, 48.2.c.a, 576.2.a.b$^{2}$

Models

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

$ 0 $ $=$ $ x^{2} + 2 x y + z^{2} $
$=$ $x^{2} + 2 x y - 5 z^{2} + w^{2} + t^{2}$
$=$ $5 x^{2} - 2 x y - 6 y^{2} - z^{2} + w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ x^{8} - 4 x^{6} y^{2} - 8 x^{6} z^{2} + 10 x^{4} y^{4} + 22 x^{4} y^{2} z^{2} + 16 x^{4} z^{4} + \cdots + 9 y^{4} z^{4} $
 Copy content Toggle raw display

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x+z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}t$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^8\,\frac{(w^{2}-wt+t^{2})(w^{2}+wt+t^{2})(1092y^{2}w^{18}+4914y^{2}w^{16}t^{2}+8532y^{2}w^{14}t^{4}+6930y^{2}w^{12}t^{6}+1674y^{2}w^{10}t^{8}-1674y^{2}w^{8}t^{10}-6930y^{2}w^{6}t^{12}-8532y^{2}w^{4}t^{14}-4914y^{2}w^{2}t^{16}-1092y^{2}t^{18}-243w^{20}-1215w^{18}t^{2}-2506w^{16}t^{4}-2734w^{14}t^{6}-1726w^{12}t^{8}-712w^{10}t^{10}-292w^{8}t^{12}-157w^{6}t^{14}-265w^{4}t^{16}-214w^{2}t^{18}-61t^{20})}{t^{4}w^{4}(w^{2}+t^{2})^{2}(24y^{2}w^{10}+60y^{2}w^{8}t^{2}+24y^{2}w^{6}t^{4}-24y^{2}w^{4}t^{6}-60y^{2}w^{2}t^{8}-24y^{2}t^{10}+w^{8}t^{4}+2w^{6}t^{6}+15w^{4}t^{8}+14w^{2}t^{10}+4t^{12})}$

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
12.96.3.f.2 $12$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.1.ci.1 $24$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.96.1.cp.1 $24$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
24.96.1.cp.2 $24$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
24.96.3.bd.1 $24$ $2$ $2$ $3$ $2$ $2$
24.96.3.cd.2 $24$ $2$ $2$ $3$ $1$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.384.17.cu.1 $24$ $2$ $2$ $17$ $3$ $1^{6}\cdot2\cdot4$
24.384.17.db.1 $24$ $2$ $2$ $17$ $3$ $1^{6}\cdot2\cdot4$
24.384.17.ne.1 $24$ $2$ $2$ $17$ $3$ $1^{6}\cdot2\cdot4$
24.384.17.nl.1 $24$ $2$ $2$ $17$ $4$ $1^{6}\cdot2\cdot4$
24.576.25.bg.1 $24$ $3$ $3$ $25$ $4$ $1^{10}\cdot2^{5}$
120.384.17.bkc.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.bke.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.bzu.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.bzw.1 $120$ $2$ $2$ $17$ $?$ not computed
168.384.17.bkc.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.bke.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.bzu.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.bzw.1 $168$ $2$ $2$ $17$ $?$ not computed
264.384.17.bkc.2 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.bke.2 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.bzu.2 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.bzw.2 $264$ $2$ $2$ $17$ $?$ not computed
312.384.17.bkc.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.bke.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.bzu.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.bzw.1 $312$ $2$ $2$ $17$ $?$ not computed