Properties

Label 24.72.1.ew.1
Level $24$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $288$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $12^{2}\cdot24^{2}$ Cusp orbits $2^{2}$
Elliptic points: $16$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.72.1.343

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}13&1\\14&1\end{bmatrix}$, $\begin{bmatrix}13&5\\10&19\end{bmatrix}$, $\begin{bmatrix}13&9\\18&17\end{bmatrix}$, $\begin{bmatrix}21&20\\20&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 24-isogeny field degree: $16$
Cyclic 24-torsion field degree: $128$
Full 24-torsion field degree: $1024$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

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

$ 0 $ $=$ $ 2 x^{2} + z w $
$=$ $6 y^{2} - 4 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 6 y^{2} z^{2} + z^{4} $
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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 embedded model to plane model:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{(16z^{6}+w^{6})^{3}}{w^{6}z^{12}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.36.0.bv.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.36.0.ch.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.36.1.gq.1 $24$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.144.9.bv.1 $24$ $2$ $2$ $9$ $2$ $1^{8}$
24.144.9.rq.1 $24$ $2$ $2$ $9$ $1$ $1^{8}$
24.144.9.yf.1 $24$ $2$ $2$ $9$ $1$ $1^{8}$
24.144.9.yr.1 $24$ $2$ $2$ $9$ $3$ $1^{8}$
24.144.9.eio.1 $24$ $2$ $2$ $9$ $1$ $1^{8}$
24.144.9.eiq.1 $24$ $2$ $2$ $9$ $2$ $1^{8}$
24.144.9.eje.1 $24$ $2$ $2$ $9$ $1$ $1^{8}$
24.144.9.ejg.1 $24$ $2$ $2$ $9$ $4$ $1^{8}$
72.216.13.nv.1 $72$ $3$ $3$ $13$ $?$ not computed
120.144.9.bgic.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgie.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgis.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgiu.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgko.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgkq.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgle.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bglg.1 $120$ $2$ $2$ $9$ $?$ not computed
168.144.9.bcew.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.bcey.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.bcfm.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.bcfo.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.bchi.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.bchk.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.bchy.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.bcia.1 $168$ $2$ $2$ $9$ $?$ not computed
264.144.9.bckw.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.bcky.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.bclm.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.bclo.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.bcni.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.bcnk.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.bcny.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.bcoa.1 $264$ $2$ $2$ $9$ $?$ not computed
312.144.9.bcfe.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.bcfg.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.bcfu.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.bcfw.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.bchq.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.bchs.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.bcig.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.bcii.1 $312$ $2$ $2$ $9$ $?$ not computed