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Properties

Label 24.72.4.ez.1
Level $24$
Index $72$
Genus $4$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $144$
Index: $72$ $\PSL_2$-index:$72$
Genus: $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $6^{4}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3$
$\overline{\Q}$-gonality: $3$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24D4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.72.4.213

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&14\\8&11\end{bmatrix}$, $\begin{bmatrix}11&9\\0&7\end{bmatrix}$, $\begin{bmatrix}13&15\\0&5\end{bmatrix}$, $\begin{bmatrix}17&5\\16&17\end{bmatrix}$, $\begin{bmatrix}21&7\\16&15\end{bmatrix}$, $\begin{bmatrix}21&11\\8&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.144.4-24.ez.1.1, 24.144.4-24.ez.1.2, 24.144.4-24.ez.1.3, 24.144.4-24.ez.1.4, 24.144.4-24.ez.1.5, 24.144.4-24.ez.1.6, 24.144.4-24.ez.1.7, 24.144.4-24.ez.1.8, 24.144.4-24.ez.1.9, 24.144.4-24.ez.1.10, 24.144.4-24.ez.1.11, 24.144.4-24.ez.1.12, 24.144.4-24.ez.1.13, 24.144.4-24.ez.1.14, 24.144.4-24.ez.1.15, 24.144.4-24.ez.1.16, 24.144.4-24.ez.1.17, 24.144.4-24.ez.1.18, 24.144.4-24.ez.1.19, 24.144.4-24.ez.1.20, 24.144.4-24.ez.1.21, 24.144.4-24.ez.1.22, 24.144.4-24.ez.1.23, 24.144.4-24.ez.1.24, 48.144.4-24.ez.1.1, 48.144.4-24.ez.1.2, 48.144.4-24.ez.1.3, 48.144.4-24.ez.1.4, 48.144.4-24.ez.1.5, 48.144.4-24.ez.1.6, 48.144.4-24.ez.1.7, 48.144.4-24.ez.1.8, 48.144.4-24.ez.1.9, 48.144.4-24.ez.1.10, 48.144.4-24.ez.1.11, 48.144.4-24.ez.1.12, 48.144.4-24.ez.1.13, 48.144.4-24.ez.1.14, 48.144.4-24.ez.1.15, 48.144.4-24.ez.1.16, 120.144.4-24.ez.1.1, 120.144.4-24.ez.1.2, 120.144.4-24.ez.1.3, 120.144.4-24.ez.1.4, 120.144.4-24.ez.1.5, 120.144.4-24.ez.1.6, 120.144.4-24.ez.1.7, 120.144.4-24.ez.1.8, 120.144.4-24.ez.1.9, 120.144.4-24.ez.1.10, 120.144.4-24.ez.1.11, 120.144.4-24.ez.1.12, 120.144.4-24.ez.1.13, 120.144.4-24.ez.1.14, 120.144.4-24.ez.1.15, 120.144.4-24.ez.1.16, 120.144.4-24.ez.1.17, 120.144.4-24.ez.1.18, 120.144.4-24.ez.1.19, 120.144.4-24.ez.1.20, 120.144.4-24.ez.1.21, 120.144.4-24.ez.1.22, 120.144.4-24.ez.1.23, 120.144.4-24.ez.1.24, 168.144.4-24.ez.1.1, 168.144.4-24.ez.1.2, 168.144.4-24.ez.1.3, 168.144.4-24.ez.1.4, 168.144.4-24.ez.1.5, 168.144.4-24.ez.1.6, 168.144.4-24.ez.1.7, 168.144.4-24.ez.1.8, 168.144.4-24.ez.1.9, 168.144.4-24.ez.1.10, 168.144.4-24.ez.1.11, 168.144.4-24.ez.1.12, 168.144.4-24.ez.1.13, 168.144.4-24.ez.1.14, 168.144.4-24.ez.1.15, 168.144.4-24.ez.1.16, 168.144.4-24.ez.1.17, 168.144.4-24.ez.1.18, 168.144.4-24.ez.1.19, 168.144.4-24.ez.1.20, 168.144.4-24.ez.1.21, 168.144.4-24.ez.1.22, 168.144.4-24.ez.1.23, 168.144.4-24.ez.1.24, 240.144.4-24.ez.1.1, 240.144.4-24.ez.1.2, 240.144.4-24.ez.1.3, 240.144.4-24.ez.1.4, 240.144.4-24.ez.1.5, 240.144.4-24.ez.1.6, 240.144.4-24.ez.1.7, 240.144.4-24.ez.1.8, 240.144.4-24.ez.1.9, 240.144.4-24.ez.1.10, 240.144.4-24.ez.1.11, 240.144.4-24.ez.1.12, 240.144.4-24.ez.1.13, 240.144.4-24.ez.1.14, 240.144.4-24.ez.1.15, 240.144.4-24.ez.1.16, 264.144.4-24.ez.1.1, 264.144.4-24.ez.1.2, 264.144.4-24.ez.1.3, 264.144.4-24.ez.1.4, 264.144.4-24.ez.1.5, 264.144.4-24.ez.1.6, 264.144.4-24.ez.1.7, 264.144.4-24.ez.1.8, 264.144.4-24.ez.1.9, 264.144.4-24.ez.1.10, 264.144.4-24.ez.1.11, 264.144.4-24.ez.1.12, 264.144.4-24.ez.1.13, 264.144.4-24.ez.1.14, 264.144.4-24.ez.1.15, 264.144.4-24.ez.1.16, 264.144.4-24.ez.1.17, 264.144.4-24.ez.1.18, 264.144.4-24.ez.1.19, 264.144.4-24.ez.1.20, 264.144.4-24.ez.1.21, 264.144.4-24.ez.1.22, 264.144.4-24.ez.1.23, 264.144.4-24.ez.1.24, 312.144.4-24.ez.1.1, 312.144.4-24.ez.1.2, 312.144.4-24.ez.1.3, 312.144.4-24.ez.1.4, 312.144.4-24.ez.1.5, 312.144.4-24.ez.1.6, 312.144.4-24.ez.1.7, 312.144.4-24.ez.1.8, 312.144.4-24.ez.1.9, 312.144.4-24.ez.1.10, 312.144.4-24.ez.1.11, 312.144.4-24.ez.1.12, 312.144.4-24.ez.1.13, 312.144.4-24.ez.1.14, 312.144.4-24.ez.1.15, 312.144.4-24.ez.1.16, 312.144.4-24.ez.1.17, 312.144.4-24.ez.1.18, 312.144.4-24.ez.1.19, 312.144.4-24.ez.1.20, 312.144.4-24.ez.1.21, 312.144.4-24.ez.1.22, 312.144.4-24.ez.1.23, 312.144.4-24.ez.1.24
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $1024$

Jacobian

Conductor: $2^{10}\cdot3^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{4}$
Newforms: 36.2.a.a$^{3}$, 144.2.a.a

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 4 y^{2} - z^{2} - z w - w^{2} $
$=$ $3 x^{3} + 2 y z w + y w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 2 x^{4} z + 4 x^{3} z^{2} + 24 x^{2} y^{3} - 3 x^{2} z^{3} - 24 x y^{3} z + x z^{4} + 6 y^{3} z^{2} + z^{5} $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:1/2:1:0)$, $(0:-1/2:1:0)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle y-\frac{1}{2}z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{2^8}{3}\cdot\frac{(z^{2}-2zw-2w^{2})^{3}(z^{2}+4zw+w^{2})^{3}}{w^{2}(2z+w)^{2}(z^{2}+zw+w^{2})^{4}}$

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
12.36.2.s.1 $12$ $2$ $2$ $2$ $0$ $1^{2}$
24.24.0.bh.1 $24$ $3$ $3$ $0$ $0$ full Jacobian
24.36.2.cj.1 $24$ $2$ $2$ $2$ $0$ $1^{2}$
24.36.2.cu.1 $24$ $2$ $2$ $2$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.144.7.ur.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.ut.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.uz.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.vb.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.xt.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.xv.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.yb.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.yd.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.8.gg.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gg.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gh.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gh.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gi.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gi.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gj.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gj.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dc.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dc.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dd.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dd.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.de.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.de.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.df.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.df.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.9.cm.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.co.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.gu.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
48.144.9.gv.1 $48$ $2$ $2$ $9$ $0$ $1^{5}$
48.144.9.jw.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.jx.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.ls.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.lu.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
72.216.16.ez.1 $72$ $3$ $3$ $16$ $?$ not computed
120.144.7.dfk.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dfm.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dfs.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dfu.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dmi.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dmk.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dmq.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dms.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.8.qk.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qk.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.ql.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.ql.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qm.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qm.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qn.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qn.2 $120$ $2$ $2$ $8$ $?$ not computed
168.144.7.cun.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.cup.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.cuv.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.cux.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.czp.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.czr.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.czx.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.czz.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.8.qc.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qc.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qd.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qd.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qe.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qe.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qf.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qf.2 $168$ $2$ $2$ $8$ $?$ not computed
240.144.8.ea.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ea.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.eb.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.eb.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ec.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ec.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ed.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ed.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.9.ia.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.ib.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.jg.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.jh.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.mm.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.mn.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.oi.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.oj.1 $240$ $2$ $2$ $9$ $?$ not computed
264.144.7.cun.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.cup.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.cuv.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.cux.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.czp.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.czr.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.czx.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.czz.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.8.qe.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qe.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qf.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qf.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qg.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qg.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qh.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qh.2 $264$ $2$ $2$ $8$ $?$ not computed
312.144.7.czt.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.czv.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dab.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dad.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dev.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dex.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dfd.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dff.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.8.qk.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qk.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.ql.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.ql.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qm.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qm.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qn.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qn.2 $312$ $2$ $2$ $8$ $?$ not computed