Invariants
Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $32$ | ||
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: | 24.72.1.73 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&17\\14&3\end{bmatrix}$, $\begin{bmatrix}7&15\\18&11\end{bmatrix}$, $\begin{bmatrix}17&16\\20&11\end{bmatrix}$, $\begin{bmatrix}23&22\\8&23\end{bmatrix}$ |
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: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y^{2} - y z + z^{2} + w^{2} $ |
$=$ | $3 x^{2} + y w + z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 3 x^{2} y z + 3 y^{2} z^{2} + 9 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{4374yz^{15}w^{2}+8019yz^{13}w^{4}-11664yz^{11}w^{6}-17739yz^{9}w^{8}+6156yz^{7}w^{10}+5022yz^{5}w^{12}+432yz^{3}w^{14}-225yzw^{16}+729z^{18}-10935z^{14}w^{4}-12879z^{12}w^{6}+10692z^{10}w^{8}+15309z^{8}w^{10}+54z^{6}w^{12}-2430z^{4}w^{14}-540z^{2}w^{16}-125w^{18}}{w^{12}(54yz^{3}w^{2}-9yzw^{4}+27z^{6}-27z^{2}w^{4}+w^{6})}$ |
Modular covers
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 |
---|---|---|---|---|---|---|
12.36.0.r.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.36.0.cc.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.36.1.gm.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.fs.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.ph.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
24.144.9.yk.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.yp.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
24.144.9.bst.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
24.144.9.bsv.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.btj.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.btl.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
72.216.13.mn.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.144.9.bfye.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bfyf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bfyu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bfyv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgaq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgar.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgbg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgbh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbuy.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbuz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbvo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbvp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbxk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbxl.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbya.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbyb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcay.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcaz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcbo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcbp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcdk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcdl.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcea.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bceb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbvg.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbvh.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbvw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbvx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbxs.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbxt.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbyi.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbyj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |