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Properties

Label 24.96.1.cl.4
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: 24.96.1.1325

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&4\\12&19\end{bmatrix}$, $\begin{bmatrix}11&16\\12&1\end{bmatrix}$, $\begin{bmatrix}13&18\\0&13\end{bmatrix}$, $\begin{bmatrix}17&18\\18&17\end{bmatrix}$, $\begin{bmatrix}19&4\\18&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.335742
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cl.4.1, 24.192.1-24.cl.4.2, 24.192.1-24.cl.4.3, 24.192.1-24.cl.4.4, 24.192.1-24.cl.4.5, 24.192.1-24.cl.4.6, 24.192.1-24.cl.4.7, 24.192.1-24.cl.4.8, 24.192.1-24.cl.4.9, 24.192.1-24.cl.4.10, 24.192.1-24.cl.4.11, 24.192.1-24.cl.4.12, 24.192.1-24.cl.4.13, 24.192.1-24.cl.4.14, 24.192.1-24.cl.4.15, 24.192.1-24.cl.4.16, 120.192.1-24.cl.4.1, 120.192.1-24.cl.4.2, 120.192.1-24.cl.4.3, 120.192.1-24.cl.4.4, 120.192.1-24.cl.4.5, 120.192.1-24.cl.4.6, 120.192.1-24.cl.4.7, 120.192.1-24.cl.4.8, 120.192.1-24.cl.4.9, 120.192.1-24.cl.4.10, 120.192.1-24.cl.4.11, 120.192.1-24.cl.4.12, 120.192.1-24.cl.4.13, 120.192.1-24.cl.4.14, 120.192.1-24.cl.4.15, 120.192.1-24.cl.4.16, 168.192.1-24.cl.4.1, 168.192.1-24.cl.4.2, 168.192.1-24.cl.4.3, 168.192.1-24.cl.4.4, 168.192.1-24.cl.4.5, 168.192.1-24.cl.4.6, 168.192.1-24.cl.4.7, 168.192.1-24.cl.4.8, 168.192.1-24.cl.4.9, 168.192.1-24.cl.4.10, 168.192.1-24.cl.4.11, 168.192.1-24.cl.4.12, 168.192.1-24.cl.4.13, 168.192.1-24.cl.4.14, 168.192.1-24.cl.4.15, 168.192.1-24.cl.4.16, 264.192.1-24.cl.4.1, 264.192.1-24.cl.4.2, 264.192.1-24.cl.4.3, 264.192.1-24.cl.4.4, 264.192.1-24.cl.4.5, 264.192.1-24.cl.4.6, 264.192.1-24.cl.4.7, 264.192.1-24.cl.4.8, 264.192.1-24.cl.4.9, 264.192.1-24.cl.4.10, 264.192.1-24.cl.4.11, 264.192.1-24.cl.4.12, 264.192.1-24.cl.4.13, 264.192.1-24.cl.4.14, 264.192.1-24.cl.4.15, 264.192.1-24.cl.4.16, 312.192.1-24.cl.4.1, 312.192.1-24.cl.4.2, 312.192.1-24.cl.4.3, 312.192.1-24.cl.4.4, 312.192.1-24.cl.4.5, 312.192.1-24.cl.4.6, 312.192.1-24.cl.4.7, 312.192.1-24.cl.4.8, 312.192.1-24.cl.4.9, 312.192.1-24.cl.4.10, 312.192.1-24.cl.4.11, 312.192.1-24.cl.4.12, 312.192.1-24.cl.4.13, 312.192.1-24.cl.4.14, 312.192.1-24.cl.4.15, 312.192.1-24.cl.4.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^6}\cdot\frac{16x^{2}y^{30}-1152x^{2}y^{28}z^{2}-15360x^{2}y^{26}z^{4}+65232896x^{2}y^{24}z^{6}+794099712x^{2}y^{22}z^{8}+298799595520x^{2}y^{20}z^{10}-20859360116736x^{2}y^{18}z^{12}-382669808467968x^{2}y^{16}z^{14}-1349795746676736x^{2}y^{14}z^{16}+3013943907844096x^{2}y^{12}z^{18}+9687023858221056x^{2}y^{10}z^{20}+3540564880392192x^{2}y^{8}z^{22}+33784693786673152x^{2}y^{6}z^{24}+10634476463849472x^{2}y^{4}z^{26}-89579411338166272x^{2}y^{2}z^{28}-51228445761339392x^{2}z^{30}+23040xy^{28}z^{3}-1855488xy^{26}z^{5}+114229248xy^{24}z^{7}-20121649152xy^{22}z^{9}-1960403009536xy^{20}z^{11}+59254593552384xy^{18}z^{13}+821900511019008xy^{16}z^{15}+1815186323275776xy^{14}z^{17}+1592187326300160xy^{12}z^{19}+14519944397979648xy^{10}z^{21}-28773119787270144xy^{8}z^{23}-140623139146039296xy^{6}z^{25}-20793963904499712xy^{4}z^{27}+89790517570699264xy^{2}z^{29}+y^{32}-784y^{30}z^{2}+44160y^{28}z^{4}-9671680y^{26}z^{6}-674406400y^{24}z^{8}+64131104768y^{22}z^{10}+6052970496000y^{20}z^{12}-1957234212864y^{18}z^{14}-164981907652608y^{16}z^{16}-1871386775650304y^{14}z^{18}-13230640312877056y^{12}z^{20}-22737883282538496y^{10}z^{22}+42208739561832448y^{8}z^{24}+109829116986916864y^{6}z^{26}+33557094879723520y^{4}z^{28}+26247541578268672y^{2}z^{30}+51509920738050048z^{32}}{z^{4}y^{4}(y^{2}+8z^{2})^{6}(8x^{2}y^{10}+1296x^{2}y^{8}z^{2}-50688x^{2}y^{6}z^{4}-829440x^{2}y^{4}z^{6}+5308416x^{2}z^{10}-56xy^{10}z+5472xy^{8}z^{3}+211968xy^{6}z^{5}-36864xy^{4}z^{7}-7962624xy^{2}z^{9}-10616832xz^{11}+y^{12}-336y^{10}z^{2}-19952y^{8}z^{4}+96768y^{6}z^{6}+2525184y^{4}z^{8}+7962624y^{2}z^{10}+5308416z^{12})}$

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.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.o.1 $24$ $2$ $2$ $0$ $0$ 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.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.f.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.g.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.h.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bw.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bx.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bz.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ca.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.d.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.d.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.l.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.x.1 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.iz.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ja.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jc.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jr.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jt.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ju.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.iz.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ja.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jc.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jd.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jq.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jr.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jt.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ju.4 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.iz.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ja.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jc.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jd.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jq.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jr.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jt.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ju.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.iz.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ja.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jc.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jd.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jq.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jr.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jt.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ju.1 $312$ $2$ $2$ $5$ $?$ not computed