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Properties

Label 24.96.1.cp.1
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: 24.96.1.1328

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&8\\6&13\end{bmatrix}$, $\begin{bmatrix}7&8\\18&17\end{bmatrix}$, $\begin{bmatrix}13&12\\12&19\end{bmatrix}$, $\begin{bmatrix}17&2\\18&1\end{bmatrix}$, $\begin{bmatrix}19&16\\6&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.335742
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cp.1.1, 24.192.1-24.cp.1.2, 24.192.1-24.cp.1.3, 24.192.1-24.cp.1.4, 24.192.1-24.cp.1.5, 24.192.1-24.cp.1.6, 24.192.1-24.cp.1.7, 24.192.1-24.cp.1.8, 24.192.1-24.cp.1.9, 24.192.1-24.cp.1.10, 24.192.1-24.cp.1.11, 24.192.1-24.cp.1.12, 24.192.1-24.cp.1.13, 24.192.1-24.cp.1.14, 24.192.1-24.cp.1.15, 24.192.1-24.cp.1.16, 120.192.1-24.cp.1.1, 120.192.1-24.cp.1.2, 120.192.1-24.cp.1.3, 120.192.1-24.cp.1.4, 120.192.1-24.cp.1.5, 120.192.1-24.cp.1.6, 120.192.1-24.cp.1.7, 120.192.1-24.cp.1.8, 120.192.1-24.cp.1.9, 120.192.1-24.cp.1.10, 120.192.1-24.cp.1.11, 120.192.1-24.cp.1.12, 120.192.1-24.cp.1.13, 120.192.1-24.cp.1.14, 120.192.1-24.cp.1.15, 120.192.1-24.cp.1.16, 168.192.1-24.cp.1.1, 168.192.1-24.cp.1.2, 168.192.1-24.cp.1.3, 168.192.1-24.cp.1.4, 168.192.1-24.cp.1.5, 168.192.1-24.cp.1.6, 168.192.1-24.cp.1.7, 168.192.1-24.cp.1.8, 168.192.1-24.cp.1.9, 168.192.1-24.cp.1.10, 168.192.1-24.cp.1.11, 168.192.1-24.cp.1.12, 168.192.1-24.cp.1.13, 168.192.1-24.cp.1.14, 168.192.1-24.cp.1.15, 168.192.1-24.cp.1.16, 264.192.1-24.cp.1.1, 264.192.1-24.cp.1.2, 264.192.1-24.cp.1.3, 264.192.1-24.cp.1.4, 264.192.1-24.cp.1.5, 264.192.1-24.cp.1.6, 264.192.1-24.cp.1.7, 264.192.1-24.cp.1.8, 264.192.1-24.cp.1.9, 264.192.1-24.cp.1.10, 264.192.1-24.cp.1.11, 264.192.1-24.cp.1.12, 264.192.1-24.cp.1.13, 264.192.1-24.cp.1.14, 264.192.1-24.cp.1.15, 264.192.1-24.cp.1.16, 312.192.1-24.cp.1.1, 312.192.1-24.cp.1.2, 312.192.1-24.cp.1.3, 312.192.1-24.cp.1.4, 312.192.1-24.cp.1.5, 312.192.1-24.cp.1.6, 312.192.1-24.cp.1.7, 312.192.1-24.cp.1.8, 312.192.1-24.cp.1.9, 312.192.1-24.cp.1.10, 312.192.1-24.cp.1.11, 312.192.1-24.cp.1.12, 312.192.1-24.cp.1.13, 312.192.1-24.cp.1.14, 312.192.1-24.cp.1.15, 312.192.1-24.cp.1.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
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 z - y z - y w $
$=$ $12 x^{2} - 12 x y + 6 y^{2} - z^{2} - 6 z w - 6 w^{2}$
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Singular plane model Singular plane model

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

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{3^2}{2^6}\cdot\frac{5804752896xy^{21}w^{2}-3869835264xy^{19}w^{4}-102550634496xy^{17}w^{6}+194781708288xy^{15}w^{8}+74386833408xy^{13}w^{10}-607707463680xy^{11}w^{12}+1049203113984xy^{9}w^{14}-1371673460736xy^{7}w^{16}+1808514514944xy^{5}w^{18}-2598428934144xy^{3}w^{20}+3925591031808xyw^{22}+241864704y^{24}-2902376448y^{22}w^{2}-12093235200y^{20}w^{4}+69334548480y^{18}w^{6}-36360327168y^{16}w^{8}-199296516096y^{14}w^{10}+495494184960y^{12}w^{12}-731040546816y^{10}w^{14}+996184977408y^{8}w^{16}-1494597353472y^{6}w^{18}+2491086200832y^{4}w^{20}-4392470691840y^{2}w^{22}+455z^{24}+21840z^{23}w+491400z^{22}w^{2}+6886864z^{21}w^{3}+67365288z^{20}w^{4}+488284032z^{19}w^{5}+2717062048z^{18}w^{6}+11867191104z^{17}w^{7}+41288225424z^{16}w^{8}+115785300224z^{15}w^{9}+265430519040z^{14}w^{10}+509146996224z^{13}w^{11}+847714821888z^{12}w^{12}+1270730287104z^{11}w^{13}+1716496929792z^{10}w^{14}+2000774002688z^{9}w^{15}+1994011056384z^{8}w^{16}+1961328709632z^{7}w^{17}+1925312817152z^{6}w^{18}+896844517376z^{5}w^{19}-25086040064z^{4}w^{20}+1602842394624z^{3}w^{21}+323130826752z^{2}w^{22}-1946488061952zw^{23}+1358954496w^{24}}{w^{6}(746496xy^{9}w^{8}-3359232xy^{7}w^{10}+4948992xy^{5}w^{12}+981504xy^{3}w^{14}-21491712xyw^{16}+46656y^{12}w^{6}-559872y^{10}w^{8}+1150848y^{8}w^{10}+1334016y^{6}w^{12}-11927232y^{4}w^{14}+38261760y^{2}w^{16}-64z^{18}-2304z^{17}w-38016z^{16}w^{2}-380544z^{15}w^{3}-2575680z^{14}w^{4}-12421632z^{13}w^{5}-43752769z^{12}w^{6}-113276568z^{11}w^{7}-213491568z^{10}w^{8}-285058912z^{9}w^{9}-257375064z^{8}w^{10}-149677632z^{7}w^{11}-60431872z^{6}w^{12}-18312064z^{5}w^{13}+7786288z^{4}w^{14}-3827328z^{3}w^{15}-8385024z^{2}w^{16}+10745856zw^{17})}$

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.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.bz.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.bq.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.br.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bs.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.bt.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bv.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bw.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.by.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bz.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.9.e.1 $24$ $3$ $3$ $9$ $2$ $1^{4}\cdot2^{2}$
72.288.9.h.3 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.p.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.bb.4 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.kl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.km.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ko.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kp.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ks.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kt.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kv.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kw.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.kl.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.km.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ko.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kp.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ks.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kt.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kv.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kw.4 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.kl.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.km.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ko.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kp.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ks.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kt.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kv.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kw.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.kl.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.km.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ko.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kp.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ks.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kt.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kv.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kw.2 $312$ $2$ $2$ $5$ $?$ not computed