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Properties

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

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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: $16$ (none of which are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{6}\cdot4$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1601

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&9\\12&23\end{bmatrix}$, $\begin{bmatrix}5&21\\0&23\end{bmatrix}$, $\begin{bmatrix}7&9\\16&19\end{bmatrix}$, $\begin{bmatrix}7&18\\16&23\end{bmatrix}$, $\begin{bmatrix}23&18\\8&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035865
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cy.2.1, 24.192.1-24.cy.2.2, 24.192.1-24.cy.2.3, 24.192.1-24.cy.2.4, 24.192.1-24.cy.2.5, 24.192.1-24.cy.2.6, 24.192.1-24.cy.2.7, 24.192.1-24.cy.2.8, 24.192.1-24.cy.2.9, 24.192.1-24.cy.2.10, 24.192.1-24.cy.2.11, 24.192.1-24.cy.2.12, 24.192.1-24.cy.2.13, 24.192.1-24.cy.2.14, 24.192.1-24.cy.2.15, 24.192.1-24.cy.2.16, 120.192.1-24.cy.2.1, 120.192.1-24.cy.2.2, 120.192.1-24.cy.2.3, 120.192.1-24.cy.2.4, 120.192.1-24.cy.2.5, 120.192.1-24.cy.2.6, 120.192.1-24.cy.2.7, 120.192.1-24.cy.2.8, 120.192.1-24.cy.2.9, 120.192.1-24.cy.2.10, 120.192.1-24.cy.2.11, 120.192.1-24.cy.2.12, 120.192.1-24.cy.2.13, 120.192.1-24.cy.2.14, 120.192.1-24.cy.2.15, 120.192.1-24.cy.2.16, 168.192.1-24.cy.2.1, 168.192.1-24.cy.2.2, 168.192.1-24.cy.2.3, 168.192.1-24.cy.2.4, 168.192.1-24.cy.2.5, 168.192.1-24.cy.2.6, 168.192.1-24.cy.2.7, 168.192.1-24.cy.2.8, 168.192.1-24.cy.2.9, 168.192.1-24.cy.2.10, 168.192.1-24.cy.2.11, 168.192.1-24.cy.2.12, 168.192.1-24.cy.2.13, 168.192.1-24.cy.2.14, 168.192.1-24.cy.2.15, 168.192.1-24.cy.2.16, 264.192.1-24.cy.2.1, 264.192.1-24.cy.2.2, 264.192.1-24.cy.2.3, 264.192.1-24.cy.2.4, 264.192.1-24.cy.2.5, 264.192.1-24.cy.2.6, 264.192.1-24.cy.2.7, 264.192.1-24.cy.2.8, 264.192.1-24.cy.2.9, 264.192.1-24.cy.2.10, 264.192.1-24.cy.2.11, 264.192.1-24.cy.2.12, 264.192.1-24.cy.2.13, 264.192.1-24.cy.2.14, 264.192.1-24.cy.2.15, 264.192.1-24.cy.2.16, 312.192.1-24.cy.2.1, 312.192.1-24.cy.2.2, 312.192.1-24.cy.2.3, 312.192.1-24.cy.2.4, 312.192.1-24.cy.2.5, 312.192.1-24.cy.2.6, 312.192.1-24.cy.2.7, 312.192.1-24.cy.2.8, 312.192.1-24.cy.2.9, 312.192.1-24.cy.2.10, 312.192.1-24.cy.2.11, 312.192.1-24.cy.2.12, 312.192.1-24.cy.2.13, 312.192.1-24.cy.2.14, 312.192.1-24.cy.2.15, 312.192.1-24.cy.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

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

Models

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

$ 0 $ $=$ $ x^{2} - x y + 3 x z + y^{2} $
$=$ $6 x^{2} - 24 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 20 x^{4} + 16 x^{3} z + 6 x^{2} y^{2} - 12 x^{2} z^{2} + 4 x z^{3} - 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle 2y$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^2\cdot3^2}\cdot\frac{863569996680089567232xz^{23}-197901474290208866304xz^{21}w^{2}+19302848509740318720xz^{19}w^{4}-1068513217389527040xz^{17}w^{6}+38277846020653056xz^{15}w^{8}-966986696097792xz^{13}w^{10}+17986301558784xz^{11}w^{12}-247368867840xz^{9}w^{14}+2526515712xz^{7}w^{16}-17777664xz^{5}w^{18}+82368xz^{3}w^{20}-144xzw^{22}+1727140002276279582720z^{24}-431784967133693214720z^{22}w^{2}+46476936745398042624z^{20}w^{4}-2863089674008657920z^{18}w^{6}+114297422693990400z^{16}w^{8}-3198979971809280z^{14}w^{10}+65911547805696z^{12}w^{12}-1015927308288z^{10}w^{14}+11666052864z^{8}w^{16}-97310592z^{6}w^{18}+546048z^{4}w^{20}-1728z^{2}w^{22}+w^{24}}{w^{2}z^{4}(644972544xz^{17}-295612416xz^{15}w^{2}+1004251364352xz^{13}w^{4}-167371681536xz^{11}w^{6}+11078156160xz^{9}w^{8}-366747264xz^{7}w^{10}+6231168xz^{5}w^{12}-48816xz^{3}w^{14}+120xzw^{16}-1289945088z^{18}+618098688z^{16}w^{2}+2008260864000z^{14}w^{4}-376558383168z^{12}w^{6}+28692398016z^{10}w^{8}-1131458544z^{8}w^{10}+24211872z^{6}w^{12}-265788z^{4}w^{14}+1212z^{2}w^{16}-w^{18})}$

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
$X_{\pm1}(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.3 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.es.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.dt.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.du.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dz.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ea.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.el.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.em.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.er.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.es.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.n.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.k.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.bv.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.ch.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.ss.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.st.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.te.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.tf.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ui.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.uj.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.uu.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.uv.2 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.ss.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.st.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.te.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.tf.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ui.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.uj.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.uu.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.uv.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.ss.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.st.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.te.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.tf.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ui.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.uj.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.uu.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.uv.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.ss.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.st.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.te.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.tf.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ui.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.uj.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.uu.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.uv.2 $312$ $2$ $2$ $5$ $?$ not computed