Invariants
Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
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: | $1$ | ||||||
$\Q$-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: | 24.36.1.22 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&11\\4&1\end{bmatrix}$, $\begin{bmatrix}5&18\\12&23\end{bmatrix}$, $\begin{bmatrix}7&23\\22&19\end{bmatrix}$, $\begin{bmatrix}17&14\\22&11\end{bmatrix}$ |
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.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - y z + y w $ |
$=$ | $x^{2} + 2 y^{2} + 2 z^{2} + 2 z w + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 6 x^{2} y z + x^{2} z^{2} + 6 y^{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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle y$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{3yz^{8}+12yz^{7}w+12yz^{6}w^{2}+180yz^{5}w^{3}+354yz^{4}w^{4}+180yz^{3}w^{5}+12yz^{2}w^{6}+12yzw^{7}+3yw^{8}-2z^{9}-36z^{8}w-126z^{7}w^{2}-150z^{6}w^{3}-186z^{5}w^{4}+186z^{4}w^{5}+150z^{3}w^{6}+126z^{2}w^{7}+36zw^{8}+2w^{9}}{(z-w)^{3}(33yz^{5}+75yz^{4}w+42yz^{3}w^{2}-42yz^{2}w^{3}-75yzw^{4}-33yw^{5}+10z^{6}+90z^{5}w+198z^{4}w^{2}+268z^{3}w^{3}+198z^{2}w^{4}+90zw^{5}+10w^{6})}$ |
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.e.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 |
24.18.0.o.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.j.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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.d.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ca.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ep.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
24.72.3.er.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
24.72.3.qy.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ra.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.rm.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
24.72.3.ro.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
72.108.5.bi.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
72.108.5.bq.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
72.324.21.y.1 | $72$ | $9$ | $9$ | $21$ | $?$ | not computed |
120.72.3.eqa.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.eqc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.eqo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.eqq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ese.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.esg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ess.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.esu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.brk.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.bxe.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.eei.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.eek.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.eew.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.eey.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.egm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ego.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.eha.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ehc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.21.bce.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
264.72.3.eei.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.eek.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.eew.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.eey.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.egm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ego.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.eha.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ehc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.eei.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.eek.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.eew.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.eey.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.egm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ego.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.eha.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ehc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |