Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $48$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24U3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.3.4828 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&0\\8&7\end{bmatrix}$, $\begin{bmatrix}11&3\\4&17\end{bmatrix}$, $\begin{bmatrix}13&15\\12&19\end{bmatrix}$, $\begin{bmatrix}13&21\\4&19\end{bmatrix}$, $\begin{bmatrix}19&6\\4&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4^2:C_2^2\times S_3$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.ed.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 48.2.a.a, 48.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} w + x y w + x z w + y z w - z w t $ |
| $=$ | $x^{2} t + x y t + x z t + y z t - z t^{2}$ | |
| $=$ | $x^{3} + x^{2} y - x z^{2} - x z t - y z^{2} + z^{2} t$ | |
| $=$ | $x^{3} + x^{2} y + x^{2} z + x y z - x z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} z + x^{5} y^{2} + 11 x^{5} z^{2} + 5 x^{4} y^{2} z + 47 x^{4} z^{3} + 8 x^{3} y^{2} z^{2} + \cdots + 36 z^{7} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{7} - 5x^{6} - 7x^{5} - 10x^{4} - 7x^{3} - 5x^{2} - x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:1:0)$, $(0:0:1:0:0)$ |
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{1}{2^2\cdot3}\cdot\frac{962938848411648xyw^{12}-13299849829646496xyw^{10}t^{2}+37379923753306032xyw^{8}t^{4}-140743781447346432xyw^{6}t^{6}+575547716747367936xyw^{4}t^{8}-2476556695720562688xyw^{2}t^{10}+11057654871309324288xyt^{12}+5447737373884416xw^{12}t-20249454795816960xw^{10}t^{3}+67130971404632064xw^{8}t^{5}-266298505222422528xw^{6}t^{7}+1121555473602969600xw^{4}t^{9}-4928741264427319296xw^{2}t^{11}+22362436495860563968xt^{13}+947892925716570y^{2}w^{12}-8810592245097435y^{2}w^{10}t^{2}+23621609252056140y^{2}w^{8}t^{4}-92784619649556000y^{2}w^{6}t^{6}+383529779124497280y^{2}w^{4}t^{8}-1662773195733030144y^{2}w^{2}t^{10}+7465703418958070784y^{2}t^{12}+5416531022315520yw^{12}t-18426607593062400yw^{10}t^{3}+60571363282255872yw^{8}t^{5}-244799573304803328yw^{6}t^{7}+1035811609122963456yw^{4}t^{9}-4566554441951477760yw^{2}t^{11}+20768107037441654784yt^{13}-106993205379072z^{14}-13374150672384000z^{13}t-593886590690918400z^{12}t^{2}-10826117393059676160z^{11}t^{3}-70608882797018873856z^{10}t^{4}-241430726625133068288z^{9}t^{5}-499088431908051222528z^{8}t^{6}-640044050574483652608z^{7}t^{7}-429255338967081418752z^{6}t^{8}+103157283754002087936z^{5}t^{9}+621896578321610833920z^{4}t^{10}+803073829116684533760z^{3}t^{11}+619691675115618435072z^{2}t^{12}+290398321981687595008zt^{13}-3188646w^{14}+4148959752191133w^{12}t^{2}-11202696382923864w^{10}t^{4}+40987004868927792w^{8}t^{6}-164945003417508096w^{6}t^{8}+701169868157260032w^{4}t^{10}-3102041882285500416w^{2}t^{12}+48591648808662020096t^{14}}{t(209952xyw^{8}t^{3}+1971216xyw^{6}t^{5}-186084864xyw^{4}t^{7}+544094208xyw^{2}t^{9}-838328320xyt^{11}-396361728xw^{4}t^{8}+704643072xw^{2}t^{10}-998244352xt^{12}-118098y^{2}w^{10}t-1266273y^{2}w^{8}t^{3}-4114476y^{2}w^{6}t^{5}-3452544y^{2}w^{4}t^{7}+112465152y^{2}w^{2}t^{9}-284864000y^{2}t^{11}+132120576yw^{2}t^{10}-455081984yt^{12}+1486016741376z^{13}+990677827584z^{12}t-1733686198272z^{11}t^{2}-1479137034240z^{10}t^{3}-213270921216z^{9}t^{4}-41278242816z^{8}t^{5}+11816534016z^{7}t^{6}-329121792z^{6}t^{7}-2208301056z^{5}t^{8}+2232926208z^{4}t^{9}-1618821120z^{3}t^{10}-867901440z^{2}t^{11}-4022337536zt^{12}-118098w^{12}t-1423737w^{10}t^{3}-5522904w^{8}t^{5}-6683472w^{6}t^{7}+48508416w^{4}t^{9}+66923008w^{2}t^{11}-959707136t^{13})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.ed.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}Y^{2}+X^{6}Z+5X^{4}Y^{2}Z+11X^{5}Z^{2}+8X^{3}Y^{2}Z^{2}+47X^{4}Z^{3}+4X^{2}Y^{2}Z^{3}+108X^{3}Z^{4}+144X^{2}Z^{5}+108XZ^{6}+36Z^{7} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.96.3.ed.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z+\frac{1}{6}t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{1}{4}z^{3}w-\frac{1}{8}z^{2}wt-\frac{1}{72}zwt^{2}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-24.bt.2.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bt.2.21 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-12.h.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.1-12.h.1.23 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.2-24.f.2.24 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.2-24.f.2.25 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.5-24.dp.2.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.dr.2.12 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.dv.1.3 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 24.384.5-24.dx.1.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.eh.1.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ej.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.en.2.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ep.2.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.9-24.ju.1.14 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 24.384.9-24.jx.2.12 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 24.384.9-24.kb.2.15 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 24.384.9-24.kc.1.15 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 24.384.9-24.kg.2.11 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 24.384.9-24.kj.1.14 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 24.384.9-24.kn.1.15 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 24.384.9-24.ko.2.13 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot4$ |
| 24.576.13-24.ic.1.10 | $24$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 120.384.5-120.sj.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.sn.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.sv.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.sz.2.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.tz.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ud.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ul.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.up.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.9-120.bek.1.22 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bel.1.31 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.ben.1.31 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.beo.1.26 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bew.2.28 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bex.1.31 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bez.1.31 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bfa.2.30 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.5-168.sj.2.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.sn.1.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.sv.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.sz.1.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.tz.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ud.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ul.2.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.up.2.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.9-168.bdz.1.23 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bea.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bec.1.24 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bed.1.30 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bel.2.22 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bem.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.beo.1.28 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bep.2.28 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.5-264.sj.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.sn.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.sv.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.sz.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.tz.2.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ud.2.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ul.2.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.up.2.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.9-264.bde.1.23 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bdf.1.24 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bdh.1.30 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bdi.1.29 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bdq.1.23 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bdr.1.24 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bdt.1.30 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bdu.1.29 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.5-312.sj.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.sn.2.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.sv.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.sz.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.tz.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ud.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ul.2.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.up.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.9-312.bek.1.29 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bel.1.14 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.ben.1.12 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.beo.1.27 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bew.2.26 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bex.1.28 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bez.1.28 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bfa.1.24 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |