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Properties

Label 24.72.3.ug.1
Level $24$
Index $72$
Genus $3$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
Index: $72$ $\PSL_2$-index:$72$
Genus: $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $6^{4}\cdot12^{4}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12D3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.72.3.1

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&0\\6&1\end{bmatrix}$, $\begin{bmatrix}7&9\\0&5\end{bmatrix}$, $\begin{bmatrix}11&15\\18&5\end{bmatrix}$, $\begin{bmatrix}11&18\\18&5\end{bmatrix}$, $\begin{bmatrix}23&6\\0&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.144.3-24.ug.1.1, 24.144.3-24.ug.1.2, 24.144.3-24.ug.1.3, 24.144.3-24.ug.1.4, 24.144.3-24.ug.1.5, 24.144.3-24.ug.1.6, 24.144.3-24.ug.1.7, 24.144.3-24.ug.1.8, 120.144.3-24.ug.1.1, 120.144.3-24.ug.1.2, 120.144.3-24.ug.1.3, 120.144.3-24.ug.1.4, 120.144.3-24.ug.1.5, 120.144.3-24.ug.1.6, 120.144.3-24.ug.1.7, 120.144.3-24.ug.1.8, 168.144.3-24.ug.1.1, 168.144.3-24.ug.1.2, 168.144.3-24.ug.1.3, 168.144.3-24.ug.1.4, 168.144.3-24.ug.1.5, 168.144.3-24.ug.1.6, 168.144.3-24.ug.1.7, 168.144.3-24.ug.1.8, 264.144.3-24.ug.1.1, 264.144.3-24.ug.1.2, 264.144.3-24.ug.1.3, 264.144.3-24.ug.1.4, 264.144.3-24.ug.1.5, 264.144.3-24.ug.1.6, 264.144.3-24.ug.1.7, 264.144.3-24.ug.1.8, 312.144.3-24.ug.1.1, 312.144.3-24.ug.1.2, 312.144.3-24.ug.1.3, 312.144.3-24.ug.1.4, 312.144.3-24.ug.1.5, 312.144.3-24.ug.1.6, 312.144.3-24.ug.1.7, 312.144.3-24.ug.1.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $1024$

Jacobian

Conductor: $2^{18}\cdot3^{4}$
Simple: no
Squarefree: no
Decomposition: $1^{3}$
Newforms: 192.2.a.d$^{2}$, 576.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

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

$ 0 $ $=$ $ x^{6} + 6 x^{5} z + 12 x^{4} y^{2} + 30 x^{4} z^{2} + 48 x^{3} y^{2} z + 80 x^{3} z^{3} + 36 x^{2} y^{4} + \cdots + 52 z^{6} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ -2x^{7} - 14x^{4} + 16x $
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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.

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

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{4}t$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle -\frac{2}{3}z^{5}-\frac{2}{3}z^{4}w+\frac{5}{6}z^{3}w^{2}-\frac{1}{4}z^{3}t^{2}+\frac{13}{6}z^{2}w^{3}-\frac{5}{6}zw^{4}+\frac{3}{4}zw^{2}t^{2}-\frac{5}{6}w^{5}-\frac{1}{2}w^{3}t^{2}$
$\displaystyle Y$ $=$ $\displaystyle 6z^{13}w^{6}t+9z^{12}w^{7}t-54z^{11}w^{8}t+\frac{9}{4}z^{11}w^{6}t^{3}-\frac{21}{2}z^{10}w^{9}t+\frac{9}{8}z^{10}w^{7}t^{3}+\frac{819}{8}z^{9}w^{10}t-\frac{351}{16}z^{9}w^{8}t^{3}+\frac{1053}{16}z^{8}w^{11}t+\frac{351}{32}z^{8}w^{9}t^{3}-\frac{2745}{16}z^{7}w^{12}t+\frac{1917}{32}z^{7}w^{10}t^{3}-\frac{999}{16}z^{6}w^{13}t-\frac{2187}{32}z^{6}w^{11}t^{3}+\frac{2187}{16}z^{5}w^{14}t-\frac{675}{32}z^{5}w^{12}t^{3}+\frac{507}{16}z^{4}w^{15}t+\frac{1647}{32}z^{4}w^{13}t^{3}-\frac{819}{16}z^{3}w^{16}t-\frac{81}{32}z^{3}w^{14}t^{3}-\frac{189}{16}z^{2}w^{17}t-\frac{477}{32}z^{2}w^{15}t^{3}+\frac{123}{16}zw^{18}t+\frac{45}{32}zw^{16}t^{3}+\frac{9}{4}w^{19}t+\frac{27}{16}w^{17}t^{3}$
$\displaystyle Z$ $=$ $\displaystyle z^{3}w^{2}-\frac{3}{2}z^{2}w^{3}+\frac{1}{2}w^{5}$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2\,\frac{160512xw^{10}+393216xw^{8}t^{2}+260736xw^{6}t^{4}-17712xw^{4}t^{6}-92730xw^{2}t^{8}-19821xt^{10}-192y^{7}t^{4}-1248y^{5}t^{6}-3840y^{3}t^{8}-236544yzw^{9}-505344yzw^{7}t^{2}-493056yzw^{5}t^{4}-250056yzw^{3}t^{6}-40380yzwt^{8}-133632yw^{10}-334848yw^{8}t^{2}-481728yw^{6}t^{4}-393084yw^{4}t^{6}-145524yw^{2}t^{8}-25029yt^{10}-233472z^{2}w^{9}-569856z^{2}w^{7}t^{2}-543744z^{2}w^{5}t^{4}-243872z^{2}w^{3}t^{6}-28944z^{2}wt^{8}+65280zw^{10}+118272zw^{8}t^{2}+216960zw^{6}t^{4}+198952zw^{4}t^{6}+76782zw^{2}t^{8}+7305zt^{10}+185088w^{11}+437376w^{9}t^{2}+496320w^{7}t^{4}+305452w^{5}t^{6}+76554w^{3}t^{8}+12300wt^{10}}{t^{6}(84xw^{4}-72xw^{2}t^{2}+21xt^{4}-48yzw^{3}+24yzwt^{2}+36yw^{4}-36yw^{2}t^{2}+21yt^{4}-48z^{2}w^{3}+32z^{2}wt^{2}+20zw^{2}t^{2}-12zt^{4}+2w^{3}t^{2}-9wt^{4})}$

Modular covers

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Cover information

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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
$X_{\mathrm{sp}}(3)$ $3$ $6$ $6$ $0$ $0$ full Jacobian
8.6.0.f.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
6.36.0.a.1 $6$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.et.1 $24$ $3$ $3$ $1$ $0$ $1^{2}$
24.36.1.fi.1 $24$ $2$ $2$ $1$ $0$ $1^{2}$
24.36.2.eu.1 $24$ $2$ $2$ $2$ $0$ $1$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.144.5.a.1 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.144.5.y.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.144.5.dn.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.144.5.dq.1 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.144.5.iu.1 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.144.5.iv.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.144.5.jb.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.144.5.jd.1 $24$ $2$ $2$ $5$ $1$ $1^{2}$
72.216.11.ez.1 $72$ $3$ $3$ $11$ $?$ not computed
72.216.11.ge.1 $72$ $3$ $3$ $11$ $?$ not computed
72.216.11.hq.1 $72$ $3$ $3$ $11$ $?$ not computed
72.216.13.en.1 $72$ $3$ $3$ $13$ $?$ not computed
72.216.15.wo.1 $72$ $3$ $3$ $15$ $?$ not computed
120.144.5.czk.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.czl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.czr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.czs.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dro.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.drp.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.drv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.drw.1 $120$ $2$ $2$ $5$ $?$ not computed
168.144.5.bpk.1 $168$ $2$ $2$ $5$ $?$ not computed
168.144.5.bpl.1 $168$ $2$ $2$ $5$ $?$ not computed
168.144.5.bpr.1 $168$ $2$ $2$ $5$ $?$ not computed
168.144.5.bps.1 $168$ $2$ $2$ $5$ $?$ not computed
168.144.5.cae.1 $168$ $2$ $2$ $5$ $?$ not computed
168.144.5.caf.1 $168$ $2$ $2$ $5$ $?$ not computed
168.144.5.cal.1 $168$ $2$ $2$ $5$ $?$ not computed
168.144.5.cam.1 $168$ $2$ $2$ $5$ $?$ not computed
264.144.5.bpk.1 $264$ $2$ $2$ $5$ $?$ not computed
264.144.5.bpl.1 $264$ $2$ $2$ $5$ $?$ not computed
264.144.5.bpr.1 $264$ $2$ $2$ $5$ $?$ not computed
264.144.5.bps.1 $264$ $2$ $2$ $5$ $?$ not computed
264.144.5.cae.1 $264$ $2$ $2$ $5$ $?$ not computed
264.144.5.caf.1 $264$ $2$ $2$ $5$ $?$ not computed
264.144.5.cal.1 $264$ $2$ $2$ $5$ $?$ not computed
264.144.5.cam.1 $264$ $2$ $2$ $5$ $?$ not computed
312.144.5.bpk.1 $312$ $2$ $2$ $5$ $?$ not computed
312.144.5.bpl.1 $312$ $2$ $2$ $5$ $?$ not computed
312.144.5.bpr.1 $312$ $2$ $2$ $5$ $?$ not computed
312.144.5.bps.1 $312$ $2$ $2$ $5$ $?$ not computed
312.144.5.cae.1 $312$ $2$ $2$ $5$ $?$ not computed
312.144.5.caf.1 $312$ $2$ $2$ $5$ $?$ not computed
312.144.5.cal.1 $312$ $2$ $2$ $5$ $?$ not computed
312.144.5.cam.1 $312$ $2$ $2$ $5$ $?$ not computed