Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
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: | 24.24.1.110 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&5\\20&9\end{bmatrix}$, $\begin{bmatrix}11&7\\6&17\end{bmatrix}$, $\begin{bmatrix}17&7\\0&19\end{bmatrix}$, $\begin{bmatrix}17&21\\16&23\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 48.48.1-24.br.1.1, 48.48.1-24.br.1.2, 48.48.1-24.br.1.3, 48.48.1-24.br.1.4, 48.48.1-24.br.1.5, 48.48.1-24.br.1.6, 48.48.1-24.br.1.7, 48.48.1-24.br.1.8, 240.48.1-24.br.1.1, 240.48.1-24.br.1.2, 240.48.1-24.br.1.3, 240.48.1-24.br.1.4, 240.48.1-24.br.1.5, 240.48.1-24.br.1.6, 240.48.1-24.br.1.7, 240.48.1-24.br.1.8 |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $3072$ |
Jacobian
Conductor: | $2^{6}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 7 x y + 5 x z - y w + z w $ |
$=$ | $25 x^{2} - 2 x w + 2 y^{2} + 2 y z + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 49 x^{4} + 119 x^{3} z + 30 x^{2} y^{2} + 144 x^{2} z^{2} + 12 x y^{2} z + 95 x z^{3} + 30 y^{2} z^{2} + 25 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^4\cdot3\,\frac{212425061250xz^{4}w-114083521200xz^{2}w^{3}+10825590384xw^{5}+213778740000y^{2}z^{4}-43873322850y^{2}z^{2}w^{2}+11820816y^{2}w^{4}+213778740000yz^{5}-60170920350yz^{3}w^{2}-6626569284yzw^{4}+49647880625z^{6}+110370833400z^{4}w^{2}-17573333184z^{2}w^{4}-1237936392w^{6}}{1513395000xz^{4}w-1914948300xz^{2}w^{3}-300710844xw^{5}-346027500y^{2}z^{4}+688193100y^{2}z^{2}w^{2}-68687731y^{2}w^{4}-346027500yz^{5}+124253100yz^{3}w^{2}+379383869yzw^{4}-346027500z^{6}+424943100z^{4}w^{2}+130693769z^{2}w^{4}+34387122w^{6}}$ |
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.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
12.12.0.o.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.0.bv.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.k.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.ce.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.fa.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.fd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.ft.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.fz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.hq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.hs.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.72.5.gh.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
24.96.5.cx.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
120.48.1.sv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.sz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.tl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.tp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.wn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.wr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.xt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.xx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.120.9.dn.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
120.144.9.gfb.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
120.240.17.bof.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
168.48.1.st.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.sx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.tj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.tn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.wl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.wp.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.xr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.xv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.13.dv.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
264.48.1.st.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.sx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.tj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.tn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.wl.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.wp.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.xr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.xv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.288.21.dv.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
312.48.1.sv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.sz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.tl.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.tp.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.wn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.wr.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.xt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.xx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |