Invariants
Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $24$ | ||
Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $4$ 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: | 12L1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.210 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&11\\4&1\end{bmatrix}$, $\begin{bmatrix}1&18\\12&11\end{bmatrix}$, $\begin{bmatrix}3&8\\4&9\end{bmatrix}$, $\begin{bmatrix}5&22\\22&13\end{bmatrix}$ |
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: | $2048$ |
Jacobian
Conductor: | $2^{3}\cdot3$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 24.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} + y w $ |
$=$ | $4 y^{2} - 2 y w + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 2 x^{2} z^{2} + 2 y^{2} z^{2} + 4 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{32yz^{8}-96yz^{6}w^{2}+216yz^{4}w^{4}-108yz^{2}w^{6}+64z^{8}w-48z^{6}w^{3}-54z^{2}w^{7}+27w^{9}}{w^{3}(24yz^{4}w+8yz^{2}w^{3}-8z^{6}+4z^{2}w^{4}+w^{6})}$ |
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 |
---|---|---|---|---|---|---|
12.18.1.j.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.18.0.c.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.0.o.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 |
---|---|---|---|---|---|---|
24.72.3.bg.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.cj.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.ex.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ez.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ut.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.uw.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.va.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.vd.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.5.bv.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
72.324.21.bl.1 | $72$ | $9$ | $9$ | $21$ | $?$ | not computed |
120.72.3.evh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.evj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.evv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.evx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.eyv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.eyx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ezj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ezl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.brx.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.bxr.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.ejp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ejr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ekd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ekf.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.end.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.enf.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.enr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ent.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.21.bcr.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
264.72.3.ejp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ejr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ekd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ekf.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.end.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.enf.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.enr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ent.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ejp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ejr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ekd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ekf.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.end.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.enf.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.enr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ent.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |