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Properties

Label 24.96.1.cl.2
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $192$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $2$ are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1337

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&8\\18&7\end{bmatrix}$, $\begin{bmatrix}5&18\\18&7\end{bmatrix}$, $\begin{bmatrix}7&16\\18&19\end{bmatrix}$, $\begin{bmatrix}23&2\\6&5\end{bmatrix}$, $\begin{bmatrix}23&16\\0&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.335742
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cl.2.1, 24.192.1-24.cl.2.2, 24.192.1-24.cl.2.3, 24.192.1-24.cl.2.4, 24.192.1-24.cl.2.5, 24.192.1-24.cl.2.6, 24.192.1-24.cl.2.7, 24.192.1-24.cl.2.8, 24.192.1-24.cl.2.9, 24.192.1-24.cl.2.10, 24.192.1-24.cl.2.11, 24.192.1-24.cl.2.12, 24.192.1-24.cl.2.13, 24.192.1-24.cl.2.14, 24.192.1-24.cl.2.15, 24.192.1-24.cl.2.16, 120.192.1-24.cl.2.1, 120.192.1-24.cl.2.2, 120.192.1-24.cl.2.3, 120.192.1-24.cl.2.4, 120.192.1-24.cl.2.5, 120.192.1-24.cl.2.6, 120.192.1-24.cl.2.7, 120.192.1-24.cl.2.8, 120.192.1-24.cl.2.9, 120.192.1-24.cl.2.10, 120.192.1-24.cl.2.11, 120.192.1-24.cl.2.12, 120.192.1-24.cl.2.13, 120.192.1-24.cl.2.14, 120.192.1-24.cl.2.15, 120.192.1-24.cl.2.16, 168.192.1-24.cl.2.1, 168.192.1-24.cl.2.2, 168.192.1-24.cl.2.3, 168.192.1-24.cl.2.4, 168.192.1-24.cl.2.5, 168.192.1-24.cl.2.6, 168.192.1-24.cl.2.7, 168.192.1-24.cl.2.8, 168.192.1-24.cl.2.9, 168.192.1-24.cl.2.10, 168.192.1-24.cl.2.11, 168.192.1-24.cl.2.12, 168.192.1-24.cl.2.13, 168.192.1-24.cl.2.14, 168.192.1-24.cl.2.15, 168.192.1-24.cl.2.16, 264.192.1-24.cl.2.1, 264.192.1-24.cl.2.2, 264.192.1-24.cl.2.3, 264.192.1-24.cl.2.4, 264.192.1-24.cl.2.5, 264.192.1-24.cl.2.6, 264.192.1-24.cl.2.7, 264.192.1-24.cl.2.8, 264.192.1-24.cl.2.9, 264.192.1-24.cl.2.10, 264.192.1-24.cl.2.11, 264.192.1-24.cl.2.12, 264.192.1-24.cl.2.13, 264.192.1-24.cl.2.14, 264.192.1-24.cl.2.15, 264.192.1-24.cl.2.16, 312.192.1-24.cl.2.1, 312.192.1-24.cl.2.2, 312.192.1-24.cl.2.3, 312.192.1-24.cl.2.4, 312.192.1-24.cl.2.5, 312.192.1-24.cl.2.6, 312.192.1-24.cl.2.7, 312.192.1-24.cl.2.8, 312.192.1-24.cl.2.9, 312.192.1-24.cl.2.10, 312.192.1-24.cl.2.11, 312.192.1-24.cl.2.12, 312.192.1-24.cl.2.13, 312.192.1-24.cl.2.14, 312.192.1-24.cl.2.15, 312.192.1-24.cl.2.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} + 3x - 3 $
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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.

Weierstrass model
$(0:1:0)$, $(1:0:1)$

Maps to other modular curves

$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^2}\cdot\frac{16x^{2}y^{30}+2821248x^{2}y^{28}z^{2}+7304033280x^{2}y^{26}z^{4}+769548886016x^{2}y^{24}z^{6}+17661894131712x^{2}y^{22}z^{8}+176318749081600x^{2}y^{20}z^{10}+4497468800630784x^{2}y^{18}z^{12}+165409340818194432x^{2}y^{16}z^{14}+2983525021818814464x^{2}y^{14}z^{16}+25632090174401806336x^{2}y^{12}z^{18}+58059595351616126976x^{2}y^{10}z^{20}-798695387899424145408x^{2}y^{8}z^{22}-7667518083839345819648x^{2}y^{6}z^{24}-28540212271055842050048x^{2}y^{4}z^{26}-47709013045832310587392x^{2}y^{2}z^{28}-27262293209257745580032x^{2}z^{30}+2880xy^{30}z+26795520xy^{28}z^{3}+45526474752xy^{26}z^{5}+1785092702208xy^{24}z^{7}+36595786579968xy^{22}z^{9}+277634481127424xy^{20}z^{11}-7302939521581056xy^{18}z^{13}-140599674129088512xy^{16}z^{15}+1565556273133387776xy^{14}z^{17}+73169293279651430400xy^{12}z^{19}+983770238716999630848xy^{10}z^{21}+6941295060535901945856xy^{8}z^{23}+27475280191614420516864xy^{6}z^{25}+57080426600397451296768xy^{4}z^{27}+47709013256938543120384xy^{2}z^{29}+y^{32}+36656y^{30}z^{2}+974901120y^{28}z^{4}+219990256640y^{26}z^{6}+6264136048640y^{24}z^{8}+71883165728768y^{22}z^{10}+261565812572160y^{20}z^{12}-25127910012616704y^{18}z^{14}-1106918490607976448y^{16}z^{16}-22428912064516849664y^{14}z^{18}-260818897549316325376y^{12}z^{20}-1845310773072972742656y^{10}z^{22}-7873096944563353812992y^{8}z^{24}-18529834003593351397376y^{6}z^{26}-16612956898535439073280y^{4}z^{28}+13631147237947570388992y^{2}z^{30}+27262293490732722290688z^{32}}{zy^{4}(y^{2}+8z^{2})^{2}(112x^{2}y^{20}z-97792x^{2}y^{18}z^{3}-16081152x^{2}y^{16}z^{5}+52953088x^{2}y^{14}z^{7}+23182639104x^{2}y^{12}z^{9}+278334013440x^{2}y^{10}z^{11}+558037467136x^{2}y^{8}z^{13}+1073741824x^{2}y^{6}z^{15}-4294967296x^{2}y^{4}z^{17}-8589934592x^{2}y^{2}z^{19}-4294967296x^{2}z^{21}-xy^{22}+1640xy^{20}z^{2}+490944xy^{18}z^{4}-37654528xy^{16}z^{6}-5075755008xy^{14}z^{8}-98772910080xy^{12}z^{10}-417796194304xy^{10}z^{12}+369098752xy^{8}z^{14}-822083584xy^{6}z^{16}-2013265920xy^{4}z^{18}-1073741824xy^{2}z^{20}-13y^{22}z-8152y^{20}z^{3}+2790336y^{18}z^{5}+381992192y^{16}z^{7}+9569214464y^{14}z^{9}+40454062080y^{12}z^{11}-139940331520y^{10}z^{13}-554690412544y^{8}z^{15}+1929379840y^{6}z^{17}-15166603264y^{4}z^{19}-26843545600y^{2}z^{21}-12884901888z^{23})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.0.a.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.o.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.bx.1 $24$ $2$ $2$ $1$ $0$ dimension zero

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.192.5.e.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.f.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.g.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.h.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bw.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bx.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bz.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ca.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.d.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.d.3 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.l.2 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.x.3 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.iz.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ja.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jc.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jq.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jr.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jt.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ju.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.iz.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ja.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jc.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jd.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jq.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jr.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jt.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ju.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.iz.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ja.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jc.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jd.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jq.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jr.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jt.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ju.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.iz.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ja.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jc.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jd.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jq.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jr.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jt.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ju.4 $312$ $2$ $2$ $5$ $?$ not computed