Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 12B2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.2.327 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&8\\20&17\end{bmatrix}$, $\begin{bmatrix}15&4\\16&15\end{bmatrix}$, $\begin{bmatrix}17&2\\2&7\end{bmatrix}$, $\begin{bmatrix}19&16\\4&17\end{bmatrix}$, $\begin{bmatrix}23&10\\20&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.36.2.a.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{8}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{2}$ |
| Newforms: | 36.2.a.a, 576.2.a.e |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} y - x z^{2} - y w^{2} $ |
| $=$ | $2 x^{2} w + 8 x y z - w^{3}$ | |
| $=$ | $8 y^{2} w + z w^{2}$ | |
| $=$ | $8 y^{2} z + z^{2} w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{3} y - 2 y^{2} z^{2} + z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{3} y $ | $=$ | $ 2 $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:1:0)$, $(1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^9\,\frac{4x^{6}z^{2}-2x^{4}z^{2}w^{2}+2x^{2}z^{2}w^{4}+xyw^{6}-8z^{8}+4z^{5}w^{3}-z^{2}w^{6}}{w^{4}z^{2}(2x^{2}-w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.36.2.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y-2Y^{2}Z^{2}+Z^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.36.2.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{32}xw^{2}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.36.1-6.a.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | $1$ |
| 24.36.1-6.a.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | $1$ |
| 24.24.0-8.a.1.1 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.3-24.a.1.3 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.c.1.6 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.g.1.5 | $24$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 24.144.3-24.i.1.2 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.z.1.10 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.ba.1.5 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.bc.1.5 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.bd.1.7 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.4-24.a.1.6 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.a.1.10 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.c.1.12 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.c.1.23 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.e.1.8 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.e.1.16 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.g.1.11 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.g.1.12 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bb.1.5 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bb.1.6 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bd.1.5 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bd.1.7 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bi.1.1 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.bi.1.6 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.bk.1.3 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bk.1.5 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 72.216.8-72.a.1.12 | $72$ | $3$ | $3$ | $8$ | $?$ | not computed |
| 120.144.3-120.bd.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.be.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bg.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bh.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cb.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cc.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ce.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cf.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.4-120.b.1.22 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.b.1.28 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.d.1.8 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.d.1.32 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.i.1.16 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.i.1.32 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.k.1.10 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.k.1.16 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.br.1.5 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.br.1.7 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bt.1.5 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bt.1.23 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.by.1.1 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.by.1.23 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ca.1.3 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ca.1.5 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.360.14-120.a.1.16 | $120$ | $5$ | $5$ | $14$ | $?$ | not computed |
| 120.432.15-120.a.1.20 | $120$ | $6$ | $6$ | $15$ | $?$ | not computed |
| 168.144.3-168.z.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.ba.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.bc.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.bd.1.5 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.bx.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.by.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.ca.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cb.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.4-168.b.1.13 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.b.1.27 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.d.1.15 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.d.1.29 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.i.1.15 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.i.1.31 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.k.1.9 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.k.1.15 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.br.1.6 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.br.1.8 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bt.1.6 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bt.1.24 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.by.1.2 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.by.1.24 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ca.1.4 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ca.1.10 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.3-264.z.1.6 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.ba.1.14 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.bc.1.14 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.bd.1.5 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.bx.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.by.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.ca.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cb.1.12 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.4-264.b.1.21 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.b.1.27 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.d.1.7 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.d.1.31 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.i.1.15 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.i.1.31 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.k.1.9 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.k.1.15 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.br.1.10 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.br.1.12 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.bt.1.10 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.bt.1.28 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.by.1.2 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.by.1.24 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ca.1.4 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ca.1.10 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.3-312.z.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.ba.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.bc.1.13 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.bd.1.3 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.bx.1.7 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.by.1.3 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.ca.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cb.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.4-312.b.1.22 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.b.1.28 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.d.1.8 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.d.1.32 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.i.1.16 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.i.1.32 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.k.1.10 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.k.1.16 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.br.1.5 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.br.1.7 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.bt.1.5 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.bt.1.23 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.by.1.1 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.by.1.23 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ca.1.3 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ca.1.5 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |