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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $288$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $2^{4}\cdot4^{2}$
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: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.271

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&8\\16&19\end{bmatrix}$, $\begin{bmatrix}5&20\\20&9\end{bmatrix}$, $\begin{bmatrix}7&16\\16&17\end{bmatrix}$, $\begin{bmatrix}17&4\\0&1\end{bmatrix}$, $\begin{bmatrix}17&16\\0&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^4\times \GL(2,3)$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.o.1.1, 24.192.1-24.o.1.2, 24.192.1-24.o.1.3, 24.192.1-24.o.1.4, 24.192.1-24.o.1.5, 24.192.1-24.o.1.6, 24.192.1-24.o.1.7, 24.192.1-24.o.1.8, 24.192.1-24.o.1.9, 24.192.1-24.o.1.10, 24.192.1-24.o.1.11, 24.192.1-24.o.1.12, 24.192.1-24.o.1.13, 24.192.1-24.o.1.14, 24.192.1-24.o.1.15, 24.192.1-24.o.1.16, 120.192.1-24.o.1.1, 120.192.1-24.o.1.2, 120.192.1-24.o.1.3, 120.192.1-24.o.1.4, 120.192.1-24.o.1.5, 120.192.1-24.o.1.6, 120.192.1-24.o.1.7, 120.192.1-24.o.1.8, 120.192.1-24.o.1.9, 120.192.1-24.o.1.10, 120.192.1-24.o.1.11, 120.192.1-24.o.1.12, 120.192.1-24.o.1.13, 120.192.1-24.o.1.14, 120.192.1-24.o.1.15, 120.192.1-24.o.1.16, 168.192.1-24.o.1.1, 168.192.1-24.o.1.2, 168.192.1-24.o.1.3, 168.192.1-24.o.1.4, 168.192.1-24.o.1.5, 168.192.1-24.o.1.6, 168.192.1-24.o.1.7, 168.192.1-24.o.1.8, 168.192.1-24.o.1.9, 168.192.1-24.o.1.10, 168.192.1-24.o.1.11, 168.192.1-24.o.1.12, 168.192.1-24.o.1.13, 168.192.1-24.o.1.14, 168.192.1-24.o.1.15, 168.192.1-24.o.1.16, 264.192.1-24.o.1.1, 264.192.1-24.o.1.2, 264.192.1-24.o.1.3, 264.192.1-24.o.1.4, 264.192.1-24.o.1.5, 264.192.1-24.o.1.6, 264.192.1-24.o.1.7, 264.192.1-24.o.1.8, 264.192.1-24.o.1.9, 264.192.1-24.o.1.10, 264.192.1-24.o.1.11, 264.192.1-24.o.1.12, 264.192.1-24.o.1.13, 264.192.1-24.o.1.14, 264.192.1-24.o.1.15, 264.192.1-24.o.1.16, 312.192.1-24.o.1.1, 312.192.1-24.o.1.2, 312.192.1-24.o.1.3, 312.192.1-24.o.1.4, 312.192.1-24.o.1.5, 312.192.1-24.o.1.6, 312.192.1-24.o.1.7, 312.192.1-24.o.1.8, 312.192.1-24.o.1.9, 312.192.1-24.o.1.10, 312.192.1-24.o.1.11, 312.192.1-24.o.1.12, 312.192.1-24.o.1.13, 312.192.1-24.o.1.14, 312.192.1-24.o.1.15, 312.192.1-24.o.1.16
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x^{2} - y^{2} - z^{2} $
$=$ $12 x y + 6 y z - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 72 x^{4} + x^{2} y^{2} + 2 x y z^{2} - z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 6z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^2}{3^2}\cdot\frac{151389735032727797760xz^{23}+4288405500412906752xz^{19}w^{4}+46481647280094720xz^{15}w^{8}+242740646698752xz^{11}w^{12}+615778417728xz^{7}w^{16}+629066736xz^{3}w^{20}+62707681454557104000y^{2}z^{22}+1688781837899599488y^{2}z^{18}w^{4}+17244624204740736y^{2}z^{14}w^{8}+83548833464448y^{2}z^{10}w^{12}+190617933096y^{2}z^{6}w^{16}+158452632y^{2}z^{2}w^{20}+4329062020330167168yz^{21}w^{2}+110543103793853568yz^{17}w^{6}+1060256075748480yz^{13}w^{10}+4742618449536yz^{9}w^{14}+9635934312yz^{5}w^{18}+5755320yzw^{22}+107048708243676463104z^{24}+3287453035641056928z^{20}w^{4}+39077365851947664z^{16}w^{8}+227931462279072z^{12}w^{12}+669248981100z^{8}w^{16}+869100246z^{4}w^{20}+389017w^{24}}{w^{8}(2811793637376xz^{15}+48249535680xz^{11}w^{4}+209679552xz^{7}w^{8}+159456xz^{3}w^{12}+1164683035872y^{2}z^{14}+18359890272y^{2}z^{10}w^{4}+67704048y^{2}z^{6}w^{8}+30672y^{2}z^{2}w^{12}+80404532064yz^{13}w^{2}+1155265632yz^{9}w^{6}+3510192yz^{5}w^{10}+720yzw^{14}+1988238348288z^{16}+38855441448z^{12}w^{4}+210696372z^{8}w^{8}+267836z^{4}w^{12}+9w^{16})}$

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
8.48.0.b.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.b.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.o.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.m.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.n.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.q.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.s.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.y.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ba.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bi.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bj.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.17.ne.2 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.eu.2 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
120.192.5.cu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.cv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.dx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.dy.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.fd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.fe.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ge.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.gf.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.cu.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.cv.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.dx.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.dy.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fd.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fe.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ge.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.gf.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.cu.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.cv.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.dx.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.dy.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.fd.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.fe.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ge.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.gf.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.cu.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.cv.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.dx.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.dy.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.fd.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.fe.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ge.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.gf.1 $312$ $2$ $2$ $5$ $?$ not computed