Invariants
Level: | $24$ | $\SL_2$-level: | $12$ | 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 | $3^{4}\cdot12^{2}$ | 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: | 12K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.136 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}17&7\\8&11\end{bmatrix}$, $\begin{bmatrix}17&7\\22&1\end{bmatrix}$, $\begin{bmatrix}17&9\\6&11\end{bmatrix}$, $\begin{bmatrix}17&18\\18&17\end{bmatrix}$ |
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: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x y + x z - y^{2} - 2 y z - z^{2} + 2 w^{2} $ |
$=$ | $2 x^{2} - x y + 3 x z + 2 y^{2} + 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 14 x^{4} + 40 x^{3} y + 44 x^{2} y^{2} - 7 x^{2} z^{2} + 24 x y^{3} - 10 x y z^{2} + 6 y^{4} - 3 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^4\cdot3^3\,\frac{192xz^{8}-592xz^{6}w^{2}-1656xz^{4}w^{4}+2148xz^{2}w^{6}+90xw^{8}+936y^{2}z^{5}w^{2}-2808y^{2}z^{3}w^{4}-558y^{2}zw^{6}-176yz^{8}+2368yz^{6}w^{2}-936yz^{4}w^{4}-1536yz^{2}w^{6}-135yw^{8}-176z^{9}+2552z^{7}w^{2}-5472z^{5}w^{4}+1686z^{3}w^{6}+1413zw^{8}}{4xz^{6}w^{2}-24xz^{2}w^{6}-9xw^{8}-18y^{2}z^{5}w^{2}+18y^{2}zw^{6}-4yz^{8}-34yz^{6}w^{2}+54yz^{4}w^{4}+60yz^{2}w^{6}-4z^{9}-8z^{7}w^{2}+54z^{5}w^{4}-30z^{3}w^{6}-36zw^{8}}$ |
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.f.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.d.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.dv.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.ip.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.it.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.mm.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.mn.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.nh.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ni.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.bw.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.bw.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.bms.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bmu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bng.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bni.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bow.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.boy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bpk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bpm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.cu.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.fm.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.bkk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bkm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bky.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bla.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bmo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bmq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bnc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bne.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.tl.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.bkk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bkm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bky.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bla.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bmo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bmq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bnc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bne.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bkk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bkm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bky.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bla.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bmo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bmq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bnc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bne.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |