Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
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: | 8C1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.98 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&5\\10&19\end{bmatrix}$, $\begin{bmatrix}11&2\\20&3\end{bmatrix}$, $\begin{bmatrix}11&4\\0&23\end{bmatrix}$, $\begin{bmatrix}23&20\\2&13\end{bmatrix}$ |
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: | $3072$ |
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 $ | $=$ | $ 3 x y - 3 x z + y w + z w $ |
$=$ | $6 x^{2} + y^{2} - y z + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 3 x^{3} z + 15 x^{2} y^{2} + 4 x^{2} z^{2} - 6 x y^{2} z - 3 x z^{3} + 15 y^{2} z^{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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{216xz^{4}w-120xz^{2}w^{3}-81y^{2}z^{4}+18y^{2}z^{2}w^{2}+7y^{2}w^{4}+81yz^{5}-18yz^{3}w^{2}-23yzw^{4}-9z^{6}-117z^{4}w^{2}-3z^{2}w^{4}+9w^{6}}{72xz^{4}w-120xz^{2}w^{3}+9y^{2}z^{4}-54y^{2}z^{2}w^{2}+y^{2}w^{4}-9yz^{5}+6yz^{3}w^{2}+55yzw^{4}+9z^{6}-39z^{4}w^{2}-9z^{2}w^{4}-9w^{6}}$ |
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 |
---|---|---|---|---|---|---|
8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
12.12.0.o.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.0.br.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.48.1.j.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.dg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.ed.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.en.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.gs.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.gu.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.hh.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.hn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.72.5.ex.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
24.96.5.ct.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
120.48.1.rx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.sb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.sn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.sr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.vp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.vt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.wf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.wj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.120.9.ct.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
120.144.9.gdr.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
120.240.17.bmf.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
168.48.1.rv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.rz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.sl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.sp.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.vn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.vr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.wd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.wh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.13.db.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
264.48.1.rv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.rz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.sl.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.sp.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.vn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.vr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.wd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.wh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.288.21.db.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
312.48.1.rx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.sb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.sn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.sr.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.vp.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.vt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.wf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.wj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |