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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
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: $1$
$\Q$-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: 24.96.1.1600

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&21\\8&11\end{bmatrix}$, $\begin{bmatrix}11&9\\12&13\end{bmatrix}$, $\begin{bmatrix}17&15\\16&23\end{bmatrix}$, $\begin{bmatrix}19&3\\8&7\end{bmatrix}$, $\begin{bmatrix}23&18\\8&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035912
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.da.2.1, 24.192.1-24.da.2.2, 24.192.1-24.da.2.3, 24.192.1-24.da.2.4, 24.192.1-24.da.2.5, 24.192.1-24.da.2.6, 24.192.1-24.da.2.7, 24.192.1-24.da.2.8, 24.192.1-24.da.2.9, 24.192.1-24.da.2.10, 24.192.1-24.da.2.11, 24.192.1-24.da.2.12, 24.192.1-24.da.2.13, 24.192.1-24.da.2.14, 24.192.1-24.da.2.15, 24.192.1-24.da.2.16, 120.192.1-24.da.2.1, 120.192.1-24.da.2.2, 120.192.1-24.da.2.3, 120.192.1-24.da.2.4, 120.192.1-24.da.2.5, 120.192.1-24.da.2.6, 120.192.1-24.da.2.7, 120.192.1-24.da.2.8, 120.192.1-24.da.2.9, 120.192.1-24.da.2.10, 120.192.1-24.da.2.11, 120.192.1-24.da.2.12, 120.192.1-24.da.2.13, 120.192.1-24.da.2.14, 120.192.1-24.da.2.15, 120.192.1-24.da.2.16, 168.192.1-24.da.2.1, 168.192.1-24.da.2.2, 168.192.1-24.da.2.3, 168.192.1-24.da.2.4, 168.192.1-24.da.2.5, 168.192.1-24.da.2.6, 168.192.1-24.da.2.7, 168.192.1-24.da.2.8, 168.192.1-24.da.2.9, 168.192.1-24.da.2.10, 168.192.1-24.da.2.11, 168.192.1-24.da.2.12, 168.192.1-24.da.2.13, 168.192.1-24.da.2.14, 168.192.1-24.da.2.15, 168.192.1-24.da.2.16, 264.192.1-24.da.2.1, 264.192.1-24.da.2.2, 264.192.1-24.da.2.3, 264.192.1-24.da.2.4, 264.192.1-24.da.2.5, 264.192.1-24.da.2.6, 264.192.1-24.da.2.7, 264.192.1-24.da.2.8, 264.192.1-24.da.2.9, 264.192.1-24.da.2.10, 264.192.1-24.da.2.11, 264.192.1-24.da.2.12, 264.192.1-24.da.2.13, 264.192.1-24.da.2.14, 264.192.1-24.da.2.15, 264.192.1-24.da.2.16, 312.192.1-24.da.2.1, 312.192.1-24.da.2.2, 312.192.1-24.da.2.3, 312.192.1-24.da.2.4, 312.192.1-24.da.2.5, 312.192.1-24.da.2.6, 312.192.1-24.da.2.7, 312.192.1-24.da.2.8, 312.192.1-24.da.2.9, 312.192.1-24.da.2.10, 312.192.1-24.da.2.11, 312.192.1-24.da.2.12, 312.192.1-24.da.2.13, 312.192.1-24.da.2.14, 312.192.1-24.da.2.15, 312.192.1-24.da.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.b

Models

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

$ 0 $ $=$ $ x^{2} - x y - 3 x z + y^{2} $
$=$ $2 x^{2} - 8 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 20 x^{4} + 16 x^{3} z + 2 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 3w$
$\displaystyle Z$ $=$ $\displaystyle 2y$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^2}\cdot\frac{1624959302500352xz^{23}-1117159614840832xz^{21}w^{2}+326895434465280xz^{19}w^{4}-54286095482880xz^{17}w^{6}+5834148151296xz^{15}w^{8}-442152124416xz^{13}w^{10}+24672567296xz^{11}w^{12}-1017978880xz^{9}w^{14}+31191552xz^{7}w^{16}-658432xz^{5}w^{18}+9152xz^{3}w^{20}-48xzw^{22}-3249918621777920z^{24}+2437438777589760z^{22}w^{2}-787091004850176z^{20}w^{4}+145460025098240z^{18}w^{6}-17420732006400z^{16}w^{8}+1462725181440z^{14}w^{10}-90413645824z^{12}w^{12}+4180770816z^{10}w^{14}-144025344z^{8}w^{16}+3604096z^{6}w^{18}-60672z^{4}w^{20}+576z^{2}w^{22}-w^{24}}{w^{2}z^{4}(32768xz^{17}-45056xz^{15}w^{2}+459191296xz^{13}w^{4}-229590784xz^{11}w^{6}+45589120xz^{9}w^{8}-4527744xz^{7}w^{10}+230784xz^{5}w^{12}-5424xz^{3}w^{14}+40xzw^{16}+65536z^{18}-94208z^{16}w^{2}-918272000z^{14}w^{4}+516540992z^{12}w^{6}-118075712z^{10}w^{8}+13968624z^{8}w^{10}-896736z^{6}w^{12}+29532z^{4}w^{14}-404z^{2}w^{16}+w^{18})}$

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.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.ik.1 $24$ $2$ $2$ $1$ $1$ 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.ex.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ey.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ez.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fa.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fn.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.fo.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fp.4 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.fq.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.9.bi.2 $24$ $3$ $3$ $9$ $2$ $1^{4}\cdot2^{2}$
72.288.9.m.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.ei.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.ej.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.wj.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wk.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wm.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wz.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xa.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xb.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xc.2 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.wj.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wk.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wl.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wm.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wz.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.xa.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.xb.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.xc.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.wj.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wk.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wl.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wm.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wz.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.xa.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.xb.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.xc.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.wj.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wk.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wl.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wm.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wz.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.xa.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.xb.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.xc.3 $312$ $2$ $2$ $5$ $?$ not computed