Invariants
Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
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: | 24.36.1.138 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&1\\22&19\end{bmatrix}$, $\begin{bmatrix}15&14\\22&15\end{bmatrix}$, $\begin{bmatrix}17&8\\10&23\end{bmatrix}$, $\begin{bmatrix}19&2\\8&17\end{bmatrix}$ |
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: | $2^{6}\cdot3$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 192.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y z $ |
$=$ | $x^{2} + 4 y^{2} - y z + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 2 x^{2} z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{54yz^{6}w^{2}-54yz^{4}w^{4}+12yz^{2}w^{6}-2yw^{8}-27z^{9}+27z^{7}w^{2}+6z^{3}w^{6}-4zw^{8}}{z^{3}(4yz^{3}w^{2}+6yzw^{4}+z^{6}+2z^{4}w^{2}-w^{6})}$ |
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 |
---|---|---|---|---|---|---|
12.18.0.b.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.0.o.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.k.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.bi.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.cf.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.fe.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ff.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.un.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.uo.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.vi.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.vj.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.5.bt.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
72.324.21.bn.1 | $72$ | $9$ | $9$ | $21$ | $?$ | not computed |
120.72.3.evn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.evp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ewb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ewd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ezb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ezd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ezp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ezr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.brz.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.bxt.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.ejv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ejx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ekj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ekl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.enj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.enl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.enx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.enz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.21.bct.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
264.72.3.ejv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ejx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ekj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ekl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.enj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.enl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.enx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.enz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ejv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ejx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ekj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ekl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.enj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.enl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.enx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.enz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |