Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| 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: | 12K3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.3.2176 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&9\\16&5\end{bmatrix}$, $\begin{bmatrix}13&21\\16&11\end{bmatrix}$, $\begin{bmatrix}17&3\\2&19\end{bmatrix}$, $\begin{bmatrix}17&9\\22&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^5.D_6$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.hi.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{16}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 48.2.a.a, 192.2.a.d, 576.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x t + x u - t^{2} - u^{2} $ |
| $=$ | $2 x z - x w + y z - w t + w u$ | |
| $=$ | $x z - 2 x w + y w - z t + z u$ | |
| $=$ | $ - y t + y u + z^{2} - z w + w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{4} y^{2} - 12 x^{2} y^{4} - 4 x^{2} y^{2} z^{2} + x^{2} z^{4} - 24 x y^{4} z - 2 x z^{5} + \cdots + z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{8} + 28x^{6} + 42x^{4} + 252x^{2} - 81 $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{4464xy^{8}u^{3}-2183904xy^{6}u^{5}+251246016xy^{4}u^{7}+4390844544xy^{2}u^{9}+144786970368xu^{11}+y^{12}-744y^{10}u^{2}+181248y^{8}u^{4}-11534528y^{6}u^{6}-673970304y^{4}u^{8}-11599689984y^{2}u^{10}-4096t^{12}+49152t^{11}u+417792t^{10}u^{2}+16912384t^{9}u^{3}-284651520t^{8}u^{4}+2139248640t^{7}u^{5}-9697687552t^{6}u^{6}+30488306688t^{5}u^{7}-70880984064t^{4}u^{8}+128084479488t^{3}u^{9}-167120561664t^{2}u^{10}+76611430656tu^{11}-108356804864u^{12}}{6xy^{8}u^{3}+240xy^{6}u^{5}+11382xy^{4}u^{7}+619668xy^{2}u^{9}+36660600xu^{11}-y^{10}u^{2}-21y^{8}u^{4}-903y^{6}u^{6}-46595y^{4}u^{8}-2662992y^{2}u^{10}-432t^{12}+5184t^{11}u-32672t^{10}u^{2}+147200t^{9}u^{3}-529776t^{8}u^{4}+1610112t^{7}u^{5}-4245120t^{6}u^{6}+9832416t^{5}u^{7}-19869712t^{4}u^{8}+34130800t^{3}u^{9}-43813008t^{2}u^{10}+20447688tu^{11}-27877032u^{12}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.hi.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 8X^{4}Y^{2}-12X^{2}Y^{4}-24XY^{4}Z-4X^{2}Y^{2}Z^{2}-12Y^{4}Z^{2}+X^{2}Z^{4}-4Y^{2}Z^{4}-2XZ^{5}+Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.96.3.hi.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t^{3}u^{2}-t^{2}u^{3}+tu^{4}-u^{5}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{8}{3}w^{3}t^{10}u^{7}-\frac{8}{3}w^{3}t^{9}u^{8}-\frac{32}{3}w^{3}t^{8}u^{9}+\frac{32}{3}w^{3}t^{7}u^{10}+48w^{3}t^{6}u^{11}-48w^{3}t^{5}u^{12}-\frac{224}{3}w^{3}t^{4}u^{13}+\frac{224}{3}w^{3}t^{3}u^{14}+\frac{104}{3}w^{3}t^{2}u^{15}-\frac{104}{3}w^{3}tu^{16}-\frac{88}{3}w^{2}t^{9}u^{9}+\frac{176}{3}w^{2}t^{8}u^{10}+\frac{80}{3}w^{2}t^{7}u^{11}-112w^{2}t^{6}u^{12}+112w^{2}t^{4}u^{14}-\frac{80}{3}w^{2}t^{3}u^{15}-\frac{176}{3}w^{2}t^{2}u^{16}+\frac{88}{3}w^{2}tu^{17}-\frac{64}{9}wt^{12}u^{7}+\frac{64}{3}wt^{11}u^{8}-\frac{464}{9}wt^{10}u^{9}+\frac{880}{9}wt^{9}u^{10}-128wt^{8}u^{11}+\frac{1280}{9}wt^{7}u^{12}-\frac{992}{9}wt^{6}u^{13}+32wt^{5}u^{14}+\frac{704}{9}wt^{4}u^{15}-\frac{1984}{9}wt^{3}u^{16}+\frac{656}{3}wt^{2}u^{17}-\frac{656}{9}wtu^{18}-\frac{16}{9}t^{11}u^{9}+\frac{64}{9}t^{10}u^{10}-\frac{32}{3}t^{9}u^{11}+\frac{64}{9}t^{8}u^{12}+\frac{32}{9}t^{7}u^{13}-\frac{64}{3}t^{6}u^{14}+\frac{832}{9}t^{5}u^{15}-\frac{2368}{9}t^{4}u^{16}+368t^{3}u^{17}-\frac{2176}{9}t^{2}u^{18}+\frac{544}{9}tu^{19}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w^{3}t^{2}+\frac{1}{2}w^{3}tu+\frac{1}{4}w^{3}u^{2}-\frac{1}{2}w^{2}tu^{2}-\frac{1}{2}w^{2}u^{3}-\frac{2}{3}wt^{4}-\frac{1}{6}wt^{2}u^{2}+\frac{5}{6}wu^{4}+\frac{1}{3}t^{3}u^{2}-\frac{1}{3}t^{2}u^{3}+\frac{2}{3}tu^{4}-\frac{2}{3}u^{5}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.1-12.o.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-12.o.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.2-24.l.1.21 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.2-24.l.1.28 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.2-24.n.1.9 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.2-24.n.1.20 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.9-24.mi.1.3 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{6}$ |
| 24.384.9-24.mo.1.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{6}$ |
| 24.384.9-24.ns.1.7 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{6}$ |
| 24.384.9-24.oc.1.6 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{6}$ |
| 24.384.9-24.pb.1.6 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{6}$ |
| 24.384.9-24.pl.1.7 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{6}$ |
| 24.384.9-24.qr.1.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{6}$ |
| 24.384.9-24.qx.1.3 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{6}$ |
| 24.576.13-24.rq.1.3 | $24$ | $3$ | $3$ | $13$ | $2$ | $1^{10}$ |
| 48.384.7-48.ih.1.6 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 48.384.7-48.ih.2.14 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 48.384.7-48.ii.1.4 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 48.384.7-48.ii.2.8 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 48.384.11-48.ek.1.14 | $48$ | $2$ | $2$ | $11$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.11-48.ek.2.6 | $48$ | $2$ | $2$ | $11$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.11-48.el.1.14 | $48$ | $2$ | $2$ | $11$ | $4$ | $1^{4}\cdot2^{2}$ |
| 48.384.11-48.el.2.6 | $48$ | $2$ | $2$ | $11$ | $4$ | $1^{4}\cdot2^{2}$ |
| 120.384.9-120.cer.1.13 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cet.1.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cff.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cfl.1.10 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cpr.1.10 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cpx.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cqj.1.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cql.1.13 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cbo.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cbq.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.ccc.1.14 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cci.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.clk.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.clq.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cmc.1.14 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cme.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.7-240.bdv.1.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bdv.2.29 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bdw.1.11 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bdw.2.15 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.11-240.fq.1.30 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.384.11-240.fq.2.6 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.384.11-240.fr.1.30 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.384.11-240.fr.2.6 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 264.384.9-264.byf.1.2 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.byh.1.2 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.byt.1.14 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.byz.1.14 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.chz.1.14 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.cif.1.14 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.cir.1.2 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.cit.1.2 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cdp.1.9 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cdr.1.7 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.ced.1.7 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cej.1.11 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cnr.1.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cnx.1.7 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.coj.1.7 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.col.1.7 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |