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: | 24.36.1.131 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&2\\8&11\end{bmatrix}$, $\begin{bmatrix}1&19\\16&23\end{bmatrix}$, $\begin{bmatrix}7&6\\6&19\end{bmatrix}$, $\begin{bmatrix}19&17\\16&13\end{bmatrix}$ |
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: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - x y + x w + 4 y^{2} - z^{2} + w^{2} $ |
$=$ | $x^{2} - 4 x y - 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 2 x^{3} y + 2 x^{2} y^{2} - 3 x^{2} z^{2} + 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{3xz^{8}-7931xz^{6}w^{2}+57438xz^{4}w^{4}+3384xz^{2}w^{6}-6360xw^{8}+12928y^{2}z^{6}w+70848y^{2}z^{4}w^{3}+206976y^{2}z^{2}w^{5}-3456y^{2}w^{7}+546yz^{8}-46764yz^{6}w^{2}-76608yz^{4}w^{4}-38160yz^{2}w^{6}+24576yw^{8}-2534z^{8}w+30928z^{6}w^{3}+47136z^{4}w^{5}+3024z^{2}w^{7}-864w^{9}}{3xz^{8}-107xz^{6}w^{2}-386xz^{4}w^{4}-216xz^{2}w^{6}+8xw^{8}-4160y^{2}z^{6}w-3392y^{2}z^{4}w^{3}-384y^{2}z^{2}w^{5}+128y^{2}w^{7}+546yz^{8}+2484yz^{6}w^{2}+288yz^{4}w^{4}+48yz^{2}w^{6}-326z^{8}w-400z^{6}w^{3}-800z^{4}w^{5}-112z^{2}w^{7}+32w^{9}}$ |
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.q.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
24.18.0.g.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.a.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.cd.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.cf.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.cy.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.db.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.dk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.dn.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.dv.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.dx.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.l.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.h.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.hx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.hz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ih.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ij.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.kt.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.kv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ld.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.lf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.l.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.x.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.hn.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.hp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.hx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.hz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.kb.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.kd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.kl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.kn.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.qg.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.hn.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.hp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.hx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.hz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.kb.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.kd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.kl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.kn.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.hn.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.hp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.hx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.hz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.kb.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.kd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.kl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.kn.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |