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: | 24.36.1.14 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&14\\22&13\end{bmatrix}$, $\begin{bmatrix}13&8\\16&13\end{bmatrix}$, $\begin{bmatrix}15&2\\10&9\end{bmatrix}$, $\begin{bmatrix}19&7\\8&7\end{bmatrix}$ |
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} + 12 x y - 2 w^{2} $ |
$=$ | $x^{2} - 3 x y - x z + 12 y^{2} + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 15 x^{4} - 6 x^{3} y + 6 x^{2} y^{2} - 9 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3}\cdot\frac{467969544xz^{8}+1090589964xz^{6}w^{2}-3822946650xz^{4}w^{4}+1480637025xz^{2}w^{6}-46282250xw^{8}+846526464y^{2}z^{7}+7201491840y^{2}z^{5}w^{2}+44719776000y^{2}z^{3}w^{4}-7985964000y^{2}zw^{6}+2520000000yz^{8}-3962973168yz^{6}w^{2}-2681544600yz^{4}w^{4}-993327300yz^{2}w^{6}-65014250yw^{8}+70543872z^{9}-1115147616z^{7}w^{2}+5244939720z^{5}w^{4}-1508286800z^{3}w^{6}-117118750zw^{8}}{2000376xz^{8}+4930956xz^{6}w^{2}-6880950xz^{4}w^{4}-5389275xz^{2}w^{6}-145250xw^{8}-10450944y^{2}z^{7}-95178240y^{2}z^{5}w^{2}-174960000y^{2}z^{3}w^{4}-144936000y^{2}zw^{6}+2258928yz^{6}w^{2}-3742200yz^{4}w^{4}+58449300yz^{2}w^{6}+9906750yw^{8}-870912z^{9}-7496064z^{7}w^{2}-9729720z^{5}w^{4}+541800z^{3}w^{6}-1963750zw^{8}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
$X_{\mathrm{ns}}(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
8.6.0.a.1 | $8$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
6.18.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.1.b.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
24.18.0.l.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.by.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.ca.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.co.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.cq.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.dg.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.di.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.do.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.dq.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.j.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.108.7.k.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.e.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.hg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.hi.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ho.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.hq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.kc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ke.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.kk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.km.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.i.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.u.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.gw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.gy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.he.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.hg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.jk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.jm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.js.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ju.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.qd.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.gw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.gy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.he.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.hg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.jk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.jm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.js.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ju.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.gw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.gy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.he.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.hg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.jk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.jm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.js.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ju.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |