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Properties

Label 24.144.5.dq.1
Level $24$
Index $144$
Genus $5$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $4$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
Index: $144$ $\PSL_2$-index:$144$
Genus: $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $4$ are rational) Cusp widths $6^{8}\cdot12^{8}$ Cusp orbits $1^{4}\cdot2^{4}\cdot4$
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: 12B5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.144.5.67

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&6\\12&11\end{bmatrix}$, $\begin{bmatrix}5&6\\12&13\end{bmatrix}$, $\begin{bmatrix}11&21\\12&5\end{bmatrix}$, $\begin{bmatrix}19&12\\0&11\end{bmatrix}$, $\begin{bmatrix}19&15\\12&13\end{bmatrix}$, $\begin{bmatrix}19&18\\12&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.5-24.dq.1.1, 24.288.5-24.dq.1.2, 24.288.5-24.dq.1.3, 24.288.5-24.dq.1.4, 24.288.5-24.dq.1.5, 24.288.5-24.dq.1.6, 24.288.5-24.dq.1.7, 24.288.5-24.dq.1.8, 24.288.5-24.dq.1.9, 24.288.5-24.dq.1.10, 24.288.5-24.dq.1.11, 24.288.5-24.dq.1.12, 24.288.5-24.dq.1.13, 24.288.5-24.dq.1.14, 120.288.5-24.dq.1.1, 120.288.5-24.dq.1.2, 120.288.5-24.dq.1.3, 120.288.5-24.dq.1.4, 120.288.5-24.dq.1.5, 120.288.5-24.dq.1.6, 120.288.5-24.dq.1.7, 120.288.5-24.dq.1.8, 120.288.5-24.dq.1.9, 120.288.5-24.dq.1.10, 120.288.5-24.dq.1.11, 120.288.5-24.dq.1.12, 120.288.5-24.dq.1.13, 120.288.5-24.dq.1.14, 168.288.5-24.dq.1.1, 168.288.5-24.dq.1.2, 168.288.5-24.dq.1.3, 168.288.5-24.dq.1.4, 168.288.5-24.dq.1.5, 168.288.5-24.dq.1.6, 168.288.5-24.dq.1.7, 168.288.5-24.dq.1.8, 168.288.5-24.dq.1.9, 168.288.5-24.dq.1.10, 168.288.5-24.dq.1.11, 168.288.5-24.dq.1.12, 168.288.5-24.dq.1.13, 168.288.5-24.dq.1.14, 264.288.5-24.dq.1.1, 264.288.5-24.dq.1.2, 264.288.5-24.dq.1.3, 264.288.5-24.dq.1.4, 264.288.5-24.dq.1.5, 264.288.5-24.dq.1.6, 264.288.5-24.dq.1.7, 264.288.5-24.dq.1.8, 264.288.5-24.dq.1.9, 264.288.5-24.dq.1.10, 264.288.5-24.dq.1.11, 264.288.5-24.dq.1.12, 264.288.5-24.dq.1.13, 264.288.5-24.dq.1.14, 312.288.5-24.dq.1.1, 312.288.5-24.dq.1.2, 312.288.5-24.dq.1.3, 312.288.5-24.dq.1.4, 312.288.5-24.dq.1.5, 312.288.5-24.dq.1.6, 312.288.5-24.dq.1.7, 312.288.5-24.dq.1.8, 312.288.5-24.dq.1.9, 312.288.5-24.dq.1.10, 312.288.5-24.dq.1.11, 312.288.5-24.dq.1.12, 312.288.5-24.dq.1.13, 312.288.5-24.dq.1.14
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $512$

Jacobian

Conductor: $2^{26}\cdot3^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{5}$
Newforms: 36.2.a.a, 192.2.a.d$^{2}$, 576.2.a.d, 576.2.a.e

Models

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

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

$ 0 $ $=$ $ - 4 x^{4} z^{4} + 24 x^{3} y^{2} z^{3} - 8 x^{3} z^{5} - 72 x^{2} y^{4} z^{2} + 56 x^{2} y^{2} z^{4} + \cdots + z^{8} $
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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
$(0:-1/2:1/2:1:0)$, $(0:-1/2:1/2:0:1)$, $(0:1/2:1/2:1:0)$, $(0:1/2:1/2:0:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

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

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 3^3\,\frac{189zw^{17}-1071zw^{16}t+3528zw^{15}t^{2}-7872zw^{14}t^{3}+13216zw^{13}t^{4}-17160zw^{12}t^{5}+17136zw^{11}t^{6}-12408zw^{10}t^{7}+4698zw^{9}t^{8}+4698zw^{8}t^{9}-12408zw^{7}t^{10}+17136zw^{6}t^{11}-17160zw^{5}t^{12}+13216zw^{4}t^{13}-7872zw^{3}t^{14}+3528zw^{2}t^{15}-1071zwt^{16}+189zt^{17}-99w^{18}+657w^{17}t-2457w^{16}t^{2}+6384w^{15}t^{3}-12614w^{14}t^{4}+19804w^{13}t^{5}-26106w^{12}t^{6}+29712w^{11}t^{7}-30660w^{10}t^{8}+31014w^{9}t^{9}-30660w^{8}t^{10}+29712w^{7}t^{11}-26106w^{6}t^{12}+19804w^{5}t^{13}-12614w^{4}t^{14}+6384w^{3}t^{15}-2457w^{2}t^{16}+657wt^{17}-99t^{18}}{t^{3}w^{3}(w^{2}-wt+t^{2})^{3}(3zw^{5}-5zw^{4}t+6zw^{3}t^{2}+6zw^{2}t^{3}-5zwt^{4}+3zt^{5}+3w^{6}-5w^{5}t+7w^{4}t^{2}-6w^{3}t^{3}+7w^{2}t^{4}-5wt^{5}+3t^{6})}$

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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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{sp}}(3)$ $3$ $12$ $12$ $0$ $0$ full Jacobian
8.12.0.k.1 $8$ $12$ $12$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.72.1.f.1 $12$ $2$ $2$ $1$ $0$ $1^{4}$
24.48.1.es.1 $24$ $3$ $3$ $1$ $0$ $1^{4}$
24.72.1.h.1 $24$ $2$ $2$ $1$ $1$ $1^{4}$
24.72.1.co.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.3.gg.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.hw.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.qg.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.ug.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.288.9.n.1 $24$ $2$ $2$ $9$ $1$ $2^{2}$
24.288.9.n.2 $24$ $2$ $2$ $9$ $1$ $2^{2}$
24.288.13.gq.1 $24$ $2$ $2$ $13$ $2$ $1^{8}$
24.288.13.hq.1 $24$ $2$ $2$ $13$ $2$ $1^{8}$
24.288.13.ii.1 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.288.13.ij.1 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.288.13.iw.1 $24$ $2$ $2$ $13$ $3$ $1^{8}$
24.288.13.jm.1 $24$ $2$ $2$ $13$ $2$ $1^{8}$
72.432.21.ib.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.iu.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.js.1 $72$ $3$ $3$ $21$ $?$ not computed
120.288.9.ct.1 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9.ct.2 $120$ $2$ $2$ $9$ $?$ not computed
120.288.13.ccg.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.ccw.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.cfk.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.cfl.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.cgo.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.che.1 $120$ $2$ $2$ $13$ $?$ not computed
168.288.9.z.1 $168$ $2$ $2$ $9$ $?$ not computed
168.288.9.z.2 $168$ $2$ $2$ $9$ $?$ not computed
168.288.13.baw.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.bbm.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.bde.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.bdf.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.bds.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.bei.1 $168$ $2$ $2$ $13$ $?$ not computed
264.288.9.z.1 $264$ $2$ $2$ $9$ $?$ not computed
264.288.9.z.2 $264$ $2$ $2$ $9$ $?$ not computed
264.288.13.zy.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.bao.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.bca.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.bcb.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.bco.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.bde.1 $264$ $2$ $2$ $13$ $?$ not computed
312.288.9.z.1 $312$ $2$ $2$ $9$ $?$ not computed
312.288.9.z.2 $312$ $2$ $2$ $9$ $?$ not computed
312.288.13.bao.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.bbe.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.bcq.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.bcr.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.bdu.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.bek.1 $312$ $2$ $2$ $13$ $?$ not computed