Invariants
Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $6^{6}$ | Cusp orbits | $2\cdot4$ | ||
Elliptic points: | $0$ 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: | 6E1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.129 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&1\\16&3\end{bmatrix}$, $\begin{bmatrix}7&11\\16&13\end{bmatrix}$, $\begin{bmatrix}17&2\\22&11\end{bmatrix}$, $\begin{bmatrix}21&13\\10&15\end{bmatrix}$ |
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: | $2048$ |
Jacobian
Conductor: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + y^{2} - y z - y w + 4 z^{2} + w^{2} $ |
$=$ | $3 x^{2} - 3 y z + y w - 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 18 x^{4} + 9 x^{2} z^{2} + 2 y^{2} z^{2} - 2 y z^{3} + 2 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle y$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^3\,\frac{16yz^{8}+224yz^{7}w+424yz^{6}w^{2}-112yz^{5}w^{3}-308yz^{4}w^{4}-88yz^{3}w^{5}-20yz^{2}w^{6}+8yzw^{7}+yw^{8}-64z^{9}+64z^{8}w+1136z^{7}w^{2}+1168z^{6}w^{3}+272z^{5}w^{4}+256z^{4}w^{5}-64z^{3}w^{6}-8z^{2}w^{7}-16zw^{8}}{z^{6}(2yz^{2}+4yzw-yw^{2}-8z^{3}+8z^{2}w-2zw^{2}+2w^{3})}$ |
Modular covers
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 |
---|---|---|---|---|---|---|
12.18.0.b.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.1.bl.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
24.18.0.k.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.e.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 |
---|---|---|---|---|---|---|
24.72.3.lh.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ll.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.mj.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.mn.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.uo.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.ur.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.vq.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.vt.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.eu.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.ex.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.ddj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ddl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ddq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dds.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.duf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.duh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dum.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.duo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.ph.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.nx.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.czf.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.czh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.czm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.czo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dmt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dmv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dna.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dnc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.bfx.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.czf.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.czh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.czm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.czo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dmt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dmv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dna.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dnc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.czf.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.czh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.czm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.czo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dmt.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dmv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dna.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dnc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |