Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.99 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&14\\14&5\end{bmatrix}$, $\begin{bmatrix}17&20\\14&7\end{bmatrix}$, $\begin{bmatrix}19&23\\12&1\end{bmatrix}$, $\begin{bmatrix}21&23\\14&7\end{bmatrix}$ |
| 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: | $3072$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x^{2} + x y - x z - y^{2} - y z - z^{2} + w^{2} $ |
| $=$ | $3 x y + 3 x z - 2 x w + y w - z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} - 2 x^{3} y + 12 x^{3} z - x^{2} y^{2} + 12 x^{2} z^{2} - 6 x y z^{2} + 18 x z^{3} - 3 y^{2} z^{2} + 9 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{249318xz^{5}-625140xz^{4}w+211440xz^{3}w^{2}-119820xz^{2}w^{3}+92640xzw^{4}-18706xw^{5}+158004y^{2}z^{4}-121752y^{2}z^{3}w-37719y^{2}z^{2}w^{2}+34722y^{2}zw^{3}-7641y^{2}w^{4}+158004yz^{5}+223956yz^{4}w-333573yz^{3}w^{2}+81159yz^{2}w^{3}-6522yzw^{4}+218yw^{5}+73629z^{6}-27900z^{5}w+2796z^{4}w^{2}-67713z^{3}w^{3}+72849z^{2}w^{4}-24530zw^{5}+3826w^{6}}{9234xz^{5}-33570xz^{4}w+59520xz^{3}w^{2}-54960xz^{2}w^{3}+22670xzw^{4}-3278xw^{5}+2727y^{2}z^{4}-8676y^{2}z^{3}w+13878y^{2}z^{2}w^{2}-9564y^{2}zw^{3}+2067y^{2}w^{4}+2727yz^{5}-5247yz^{4}w+2226yz^{3}w^{2}+6942yz^{2}w^{3}-9661yzw^{4}+2509yw^{5}+2727z^{6}-8325z^{5}w+12948z^{4}w^{2}-8394z^{3}w^{3}-1013z^{2}w^{4}+3635zw^{5}-662w^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.12.0.o.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.bu.1 | $24$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.48.1.j.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.de.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ev.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fj.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gf.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hw.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ic.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.gj.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.cz.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 120.48.1.tb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.xz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.yd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.dp.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.gfd.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.boh.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.sz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.td.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.tp.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.tt.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.wr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.wv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.xx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.yb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.dx.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.sz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.td.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.tp.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.tt.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.wr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.wv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.xx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.yb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.dx.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.tb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.tf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.tr.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.tv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.wt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.wx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.xz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.yd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |