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Properties

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

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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 $2^{4}\cdot4^{4}$
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: 24.192.5.317

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&22\\6&17\end{bmatrix}$, $\begin{bmatrix}11&0\\0&17\end{bmatrix}$, $\begin{bmatrix}11&8\\12&13\end{bmatrix}$, $\begin{bmatrix}11&20\\0&7\end{bmatrix}$, $\begin{bmatrix}17&2\\18&11\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^5.D_6$
Contains $-I$: yes
Quadratic refinements: 24.384.5-24.bq.1.1, 24.384.5-24.bq.1.2, 24.384.5-24.bq.1.3, 24.384.5-24.bq.1.4, 24.384.5-24.bq.1.5, 24.384.5-24.bq.1.6, 24.384.5-24.bq.1.7, 24.384.5-24.bq.1.8, 24.384.5-24.bq.1.9, 24.384.5-24.bq.1.10, 24.384.5-24.bq.1.11, 24.384.5-24.bq.1.12, 120.384.5-24.bq.1.1, 120.384.5-24.bq.1.2, 120.384.5-24.bq.1.3, 120.384.5-24.bq.1.4, 120.384.5-24.bq.1.5, 120.384.5-24.bq.1.6, 120.384.5-24.bq.1.7, 120.384.5-24.bq.1.8, 120.384.5-24.bq.1.9, 120.384.5-24.bq.1.10, 120.384.5-24.bq.1.11, 120.384.5-24.bq.1.12, 168.384.5-24.bq.1.1, 168.384.5-24.bq.1.2, 168.384.5-24.bq.1.3, 168.384.5-24.bq.1.4, 168.384.5-24.bq.1.5, 168.384.5-24.bq.1.6, 168.384.5-24.bq.1.7, 168.384.5-24.bq.1.8, 168.384.5-24.bq.1.9, 168.384.5-24.bq.1.10, 168.384.5-24.bq.1.11, 168.384.5-24.bq.1.12, 264.384.5-24.bq.1.1, 264.384.5-24.bq.1.2, 264.384.5-24.bq.1.3, 264.384.5-24.bq.1.4, 264.384.5-24.bq.1.5, 264.384.5-24.bq.1.6, 264.384.5-24.bq.1.7, 264.384.5-24.bq.1.8, 264.384.5-24.bq.1.9, 264.384.5-24.bq.1.10, 264.384.5-24.bq.1.11, 264.384.5-24.bq.1.12, 312.384.5-24.bq.1.1, 312.384.5-24.bq.1.2, 312.384.5-24.bq.1.3, 312.384.5-24.bq.1.4, 312.384.5-24.bq.1.5, 312.384.5-24.bq.1.6, 312.384.5-24.bq.1.7, 312.384.5-24.bq.1.8, 312.384.5-24.bq.1.9, 312.384.5-24.bq.1.10, 312.384.5-24.bq.1.11, 312.384.5-24.bq.1.12
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{28}\cdot3^{7}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2$
Newforms: 48.2.a.a, 192.2.c.a, 576.2.a.b$^{2}$

Models

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

$ 0 $ $=$ $ y^{2} - y t + w^{2} + w t $
$=$ $y t + 2 z^{2} + 2 z t - w t$
$=$ $6 x^{2} - 2 y w + y t - w t - t^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 36 x^{4} y^{4} - 144 x^{4} y^{3} z + 216 x^{4} y^{2} z^{2} - 144 x^{4} y z^{3} + 36 x^{4} z^{4} + \cdots + 9 z^{8} $
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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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{24576yw^{22}t+135168yw^{21}t^{2}+163840yw^{20}t^{3}-389120yw^{19}t^{4}-1183744yw^{18}t^{5}-718848yw^{17}t^{6}+847872yw^{16}t^{7}+1179648yw^{15}t^{8}-46080yw^{14}t^{9}-634368yw^{13}t^{10}-49152yw^{12}t^{11}+300544yw^{11}t^{12}+58624yw^{10}t^{13}-87808yw^{9}t^{14}-12288yw^{8}t^{15}+24576yw^{7}t^{16}+1632yw^{6}t^{17}-4208yw^{5}t^{18}+256yw^{4}t^{19}+464yw^{3}t^{20}-192yw^{2}t^{21}+24ywt^{22}-4096w^{24}-24576w^{23}t+286720w^{21}t^{3}+729088w^{20}t^{4}+446464w^{19}t^{5}-811008w^{18}t^{6}-1437696w^{17}t^{7}-430848w^{16}t^{8}+619520w^{15}t^{9}+344064w^{14}t^{10}-285696w^{13}t^{11}-233216w^{12}t^{12}+57088w^{11}t^{13}+67072w^{10}t^{14}-21504w^{9}t^{15}-20784w^{8}t^{16}+2976w^{7}t^{17}+2304w^{6}t^{18}-1184w^{5}t^{19}-272w^{4}t^{20}+192w^{3}t^{21}-t^{24}}{t^{6}w^{6}(w+t)^{4}(8yw^{7}-80yw^{5}t^{2}+20yw^{4}t^{3}+94yw^{3}t^{4}-64yw^{2}t^{5}+14ywt^{6}-yt^{7}+32w^{7}t+15w^{6}t^{2}-80w^{5}t^{3}-10w^{4}t^{4}+40w^{3}t^{5}-12w^{2}t^{6}+wt^{7})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.96.1.a.1 $12$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
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.bs.2 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.cc.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$

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.cv.2 $24$ $2$ $2$ $17$ $3$ $1^{6}\cdot2\cdot4$
24.384.17.da.2 $24$ $2$ $2$ $17$ $3$ $1^{6}\cdot2\cdot4$
24.384.17.nf.2 $24$ $2$ $2$ $17$ $3$ $1^{6}\cdot2\cdot4$
24.384.17.nk.2 $24$ $2$ $2$ $17$ $4$ $1^{6}\cdot2\cdot4$
24.576.25.bh.1 $24$ $3$ $3$ $25$ $4$ $1^{10}\cdot2^{5}$
120.384.17.bjz.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.bkb.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.bzr.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.bzt.1 $120$ $2$ $2$ $17$ $?$ not computed
168.384.17.bjz.2 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.bkb.2 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.bzr.2 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.bzt.2 $168$ $2$ $2$ $17$ $?$ not computed
264.384.17.bjz.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.bkb.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.bzr.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.bzt.1 $264$ $2$ $2$ $17$ $?$ not computed
312.384.17.bjz.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.bkb.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.bzr.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.bzt.2 $312$ $2$ $2$ $17$ $?$ not computed