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Properties

Label 24.192.5.bv.3
Level $24$
Index $192$
Genus $5$
Analytic rank $1$
Cusps $24$
$\Q$-cusps $4$

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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: $24$ (of which $4$ are rational) Cusp widths $4^{12}\cdot12^{12}$ Cusp orbits $1^{4}\cdot2^{2}\cdot4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12E5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.192.5.302

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&14\\12&19\end{bmatrix}$, $\begin{bmatrix}13&16\\12&17\end{bmatrix}$, $\begin{bmatrix}13&20\\12&13\end{bmatrix}$, $\begin{bmatrix}17&0\\12&7\end{bmatrix}$, $\begin{bmatrix}17&12\\0&11\end{bmatrix}$, $\begin{bmatrix}19&12\\12&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_{12}:C_2^5$
Contains $-I$: yes
Quadratic refinements: 24.384.5-24.bv.3.1, 24.384.5-24.bv.3.2, 24.384.5-24.bv.3.3, 24.384.5-24.bv.3.4, 24.384.5-24.bv.3.5, 24.384.5-24.bv.3.6, 24.384.5-24.bv.3.7, 24.384.5-24.bv.3.8, 24.384.5-24.bv.3.9, 24.384.5-24.bv.3.10, 24.384.5-24.bv.3.11, 24.384.5-24.bv.3.12, 24.384.5-24.bv.3.13, 24.384.5-24.bv.3.14, 24.384.5-24.bv.3.15, 24.384.5-24.bv.3.16, 24.384.5-24.bv.3.17, 24.384.5-24.bv.3.18, 24.384.5-24.bv.3.19, 24.384.5-24.bv.3.20, 120.384.5-24.bv.3.1, 120.384.5-24.bv.3.2, 120.384.5-24.bv.3.3, 120.384.5-24.bv.3.4, 120.384.5-24.bv.3.5, 120.384.5-24.bv.3.6, 120.384.5-24.bv.3.7, 120.384.5-24.bv.3.8, 120.384.5-24.bv.3.9, 120.384.5-24.bv.3.10, 120.384.5-24.bv.3.11, 120.384.5-24.bv.3.12, 120.384.5-24.bv.3.13, 120.384.5-24.bv.3.14, 120.384.5-24.bv.3.15, 120.384.5-24.bv.3.16, 120.384.5-24.bv.3.17, 120.384.5-24.bv.3.18, 120.384.5-24.bv.3.19, 120.384.5-24.bv.3.20, 168.384.5-24.bv.3.1, 168.384.5-24.bv.3.2, 168.384.5-24.bv.3.3, 168.384.5-24.bv.3.4, 168.384.5-24.bv.3.5, 168.384.5-24.bv.3.6, 168.384.5-24.bv.3.7, 168.384.5-24.bv.3.8, 168.384.5-24.bv.3.9, 168.384.5-24.bv.3.10, 168.384.5-24.bv.3.11, 168.384.5-24.bv.3.12, 168.384.5-24.bv.3.13, 168.384.5-24.bv.3.14, 168.384.5-24.bv.3.15, 168.384.5-24.bv.3.16, 168.384.5-24.bv.3.17, 168.384.5-24.bv.3.18, 168.384.5-24.bv.3.19, 168.384.5-24.bv.3.20, 264.384.5-24.bv.3.1, 264.384.5-24.bv.3.2, 264.384.5-24.bv.3.3, 264.384.5-24.bv.3.4, 264.384.5-24.bv.3.5, 264.384.5-24.bv.3.6, 264.384.5-24.bv.3.7, 264.384.5-24.bv.3.8, 264.384.5-24.bv.3.9, 264.384.5-24.bv.3.10, 264.384.5-24.bv.3.11, 264.384.5-24.bv.3.12, 264.384.5-24.bv.3.13, 264.384.5-24.bv.3.14, 264.384.5-24.bv.3.15, 264.384.5-24.bv.3.16, 264.384.5-24.bv.3.17, 264.384.5-24.bv.3.18, 264.384.5-24.bv.3.19, 264.384.5-24.bv.3.20, 312.384.5-24.bv.3.1, 312.384.5-24.bv.3.2, 312.384.5-24.bv.3.3, 312.384.5-24.bv.3.4, 312.384.5-24.bv.3.5, 312.384.5-24.bv.3.6, 312.384.5-24.bv.3.7, 312.384.5-24.bv.3.8, 312.384.5-24.bv.3.9, 312.384.5-24.bv.3.10, 312.384.5-24.bv.3.11, 312.384.5-24.bv.3.12, 312.384.5-24.bv.3.13, 312.384.5-24.bv.3.14, 312.384.5-24.bv.3.15, 312.384.5-24.bv.3.16, 312.384.5-24.bv.3.17, 312.384.5-24.bv.3.18, 312.384.5-24.bv.3.19, 312.384.5-24.bv.3.20
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{27}\cdot3^{7}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2$
Newforms: 24.2.a.a, 192.2.c.a, 576.2.a.b, 576.2.a.d

Models

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

$ 0 $ $=$ $ y^{2} - z^{2} + w^{2} - t^{2} $
$=$ $y^{2} + y z + y t + z w + w^{2} + w t$
$=$ $3 x^{2} - y w + z t$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 36 x^{4} y^{2} - 36 x^{4} y z - 4 y^{4} z^{2} + 8 y^{3} z^{3} - 9 y^{2} z^{4} + 5 y z^{5} - z^{6} $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:-1:1)$, $(0:-1:1:0:0)$, $(0:0:-1:1:0)$, $(0:-1:0:0:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

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

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 -\frac{4096yw^{21}t^{2}-36864yw^{20}t^{3}-163840yw^{19}t^{4}+106496yw^{18}t^{5}+1544192yw^{17}t^{6}+2609152yw^{16}t^{7}-2818048yw^{15}t^{8}-20234240yw^{14}t^{9}-36990976yw^{13}t^{10}-2641920yw^{12}t^{11}+156139520yw^{11}t^{12}+437313536yw^{10}t^{13}+577425408yw^{9}t^{14}-102707200yw^{8}t^{15}-2526248960yw^{7}t^{16}-6959792128yw^{6}t^{17}-11159224320yw^{5}t^{18}-8051408896yw^{4}t^{19}+15198650368yw^{3}t^{20}+72910635008yw^{2}t^{21}+164087570432ywt^{22}+225130504192yt^{23}-z^{24}+12z^{22}t^{2}-48z^{21}t^{3}+78z^{20}t^{4}+144z^{19}t^{5}-2084z^{18}t^{6}+9216z^{17}t^{7}-21807z^{16}t^{8}+4736z^{15}t^{9}+215064z^{14}t^{10}-1148640z^{13}t^{11}+3586500z^{12}t^{12}-6407904z^{11}t^{13}-3618792z^{10}t^{14}+82268800z^{9}t^{15}-388715823z^{8}t^{16}+1227285504z^{7}t^{17}-2758248484z^{6}t^{18}+3291365520z^{5}t^{19}+6589907022z^{4}t^{20}-58891321392z^{3}t^{21}+234096918540z^{2}t^{22}-4096zw^{23}-4096zw^{22}t+40960zw^{21}t^{2}+98304zw^{20}t^{3}-118784zw^{19}t^{4}-929792zw^{18}t^{5}-1409024zw^{17}t^{6}+2195456zw^{16}t^{7}+13619200zw^{15}t^{8}+23875584zw^{14}t^{9}-4366336zw^{13}t^{10}-125272064zw^{12}t^{11}-330469376zw^{11}t^{12}-397684736zw^{10}t^{13}+229392384zw^{9}t^{14}+2271674368zw^{8}t^{15}+5801123840zw^{7}t^{16}+8605523968zw^{6}t^{17}+4323966976zw^{5}t^{18}-17252810752zw^{4}t^{19}-61737172992zw^{3}t^{20}-100837588992zw^{2}t^{21}-61042933760zt^{23}-4096w^{24}-4096w^{23}t+40960w^{22}t^{2}+86016w^{21}t^{3}-122880w^{20}t^{4}-716800w^{19}t^{5}-843776w^{18}t^{6}+1921024w^{17}t^{7}+9150464w^{16}t^{8}+13406208w^{15}t^{9}-9068544w^{14}t^{10}-84553728w^{13}t^{11}-188702720w^{12}t^{12}-169078784w^{11}t^{13}+279257088w^{10}t^{14}+1445761024w^{9}t^{15}+3108499456w^{8}t^{16}+3734773760w^{7}t^{17}-207323136w^{6}t^{18}-13766180864w^{5}t^{19}-39742423040w^{4}t^{20}-67027369984w^{3}t^{21}-39100416000w^{2}t^{22}+225130504192wt^{23}+225130504191t^{24}}{t^{6}(384yw^{10}t^{7}+2944yw^{9}t^{8}+7168yw^{8}t^{9}-15616yw^{7}t^{10}-172672yw^{6}t^{11}-591056yw^{5}t^{12}-779568yw^{4}t^{13}+2329216yw^{3}t^{14}+17290336yw^{2}t^{15}+55782352ywt^{16}+102157904yt^{17}+z^{18}-6z^{17}t+15z^{16}t^{2}+10z^{15}t^{3}-261z^{14}t^{4}+1308z^{13}t^{5}-4214z^{12}t^{6}+9372z^{11}t^{7}-10245z^{10}t^{8}-28022z^{9}t^{9}+228111z^{8}t^{10}-945798z^{7}t^{11}+3092225z^{6}t^{12}-8797056z^{5}t^{13}+22691328z^{4}t^{14}-54262912z^{3}t^{15}+121991424z^{2}t^{16}-64zw^{11}t^{6}-448zw^{10}t^{7}+256zw^{9}t^{8}+14848zw^{8}t^{9}+76944zw^{7}t^{10}+159888zw^{6}t^{11}-239520zw^{5}t^{12}-3006848zw^{4}t^{13}-11497968zw^{3}t^{14}-23447952zw^{2}t^{15}-46375552zt^{17}-64w^{12}t^{6}-448w^{11}t^{7}-128w^{10}t^{8}+11008w^{9}t^{9}+61072w^{8}t^{10}+143120w^{7}t^{11}-71456w^{6}t^{12}-1886736w^{5}t^{13}-7880096w^{4}t^{14}-18107600w^{3}t^{15}-11949984w^{2}t^{16}+102157904wt^{17}+102157904t^{18})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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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.b.3 $12$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.96.1.cn.2 $24$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.96.1.cp.2 $24$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
24.96.3.bf.1 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.bv.2 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.bz.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
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
24.384.13.bo.2 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.384.13.bp.2 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.384.13.bw.4 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.384.13.bw.8 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.384.13.bx.4 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.384.13.bx.8 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.384.13.ce.3 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.384.13.cf.2 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.384.17.fe.4 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2\cdot4$
24.384.17.gz.4 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2\cdot4$
24.384.17.nt.4 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2\cdot4$
24.384.17.oe.4 $24$ $2$ $2$ $17$ $3$ $1^{6}\cdot2\cdot4$
24.576.25.br.1 $24$ $3$ $3$ $25$ $3$ $1^{10}\cdot2^{5}$
120.384.13.eu.2 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.ev.4 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.fo.1 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.fo.2 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.fp.1 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.fp.3 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.gi.4 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.gj.3 $120$ $2$ $2$ $13$ $?$ not computed
120.384.17.bmf.2 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.bmj.2 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.cbx.2 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.ccb.2 $120$ $2$ $2$ $17$ $?$ not computed
168.384.13.eu.4 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.ev.3 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.fo.5 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.fo.7 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.fp.5 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.fp.7 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.gi.3 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.gj.3 $168$ $2$ $2$ $13$ $?$ not computed
168.384.17.bmf.4 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.bmj.4 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.cbx.4 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.ccb.4 $168$ $2$ $2$ $17$ $?$ not computed
264.384.13.eu.3 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.ev.4 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.fo.3 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.fo.4 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.fp.3 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.fp.6 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.gi.2 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.gj.4 $264$ $2$ $2$ $13$ $?$ not computed
264.384.17.bmf.4 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.bmj.4 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.cbx.4 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.ccb.4 $264$ $2$ $2$ $17$ $?$ not computed
312.384.13.eu.2 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.ev.2 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.fo.1 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.fo.3 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.fp.3 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.fp.4 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.gi.2 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.gj.2 $312$ $2$ $2$ $13$ $?$ not computed
312.384.17.bmf.4 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.bmj.4 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.cbx.4 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.ccb.4 $312$ $2$ $2$ $17$ $?$ not computed