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Properties

Label 24.96.1.dc.4
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: $576$
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: $0$
$\Q$-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: 24.96.1.1613

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&6\\4&11\end{bmatrix}$, $\begin{bmatrix}1&12\\4&5\end{bmatrix}$, $\begin{bmatrix}13&15\\16&1\end{bmatrix}$, $\begin{bmatrix}13&18\\4&19\end{bmatrix}$, $\begin{bmatrix}23&0\\8&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035916
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dc.4.1, 24.192.1-24.dc.4.2, 24.192.1-24.dc.4.3, 24.192.1-24.dc.4.4, 24.192.1-24.dc.4.5, 24.192.1-24.dc.4.6, 24.192.1-24.dc.4.7, 24.192.1-24.dc.4.8, 24.192.1-24.dc.4.9, 24.192.1-24.dc.4.10, 24.192.1-24.dc.4.11, 24.192.1-24.dc.4.12, 24.192.1-24.dc.4.13, 24.192.1-24.dc.4.14, 24.192.1-24.dc.4.15, 24.192.1-24.dc.4.16, 120.192.1-24.dc.4.1, 120.192.1-24.dc.4.2, 120.192.1-24.dc.4.3, 120.192.1-24.dc.4.4, 120.192.1-24.dc.4.5, 120.192.1-24.dc.4.6, 120.192.1-24.dc.4.7, 120.192.1-24.dc.4.8, 120.192.1-24.dc.4.9, 120.192.1-24.dc.4.10, 120.192.1-24.dc.4.11, 120.192.1-24.dc.4.12, 120.192.1-24.dc.4.13, 120.192.1-24.dc.4.14, 120.192.1-24.dc.4.15, 120.192.1-24.dc.4.16, 168.192.1-24.dc.4.1, 168.192.1-24.dc.4.2, 168.192.1-24.dc.4.3, 168.192.1-24.dc.4.4, 168.192.1-24.dc.4.5, 168.192.1-24.dc.4.6, 168.192.1-24.dc.4.7, 168.192.1-24.dc.4.8, 168.192.1-24.dc.4.9, 168.192.1-24.dc.4.10, 168.192.1-24.dc.4.11, 168.192.1-24.dc.4.12, 168.192.1-24.dc.4.13, 168.192.1-24.dc.4.14, 168.192.1-24.dc.4.15, 168.192.1-24.dc.4.16, 264.192.1-24.dc.4.1, 264.192.1-24.dc.4.2, 264.192.1-24.dc.4.3, 264.192.1-24.dc.4.4, 264.192.1-24.dc.4.5, 264.192.1-24.dc.4.6, 264.192.1-24.dc.4.7, 264.192.1-24.dc.4.8, 264.192.1-24.dc.4.9, 264.192.1-24.dc.4.10, 264.192.1-24.dc.4.11, 264.192.1-24.dc.4.12, 264.192.1-24.dc.4.13, 264.192.1-24.dc.4.14, 264.192.1-24.dc.4.15, 264.192.1-24.dc.4.16, 312.192.1-24.dc.4.1, 312.192.1-24.dc.4.2, 312.192.1-24.dc.4.3, 312.192.1-24.dc.4.4, 312.192.1-24.dc.4.5, 312.192.1-24.dc.4.6, 312.192.1-24.dc.4.7, 312.192.1-24.dc.4.8, 312.192.1-24.dc.4.9, 312.192.1-24.dc.4.10, 312.192.1-24.dc.4.11, 312.192.1-24.dc.4.12, 312.192.1-24.dc.4.13, 312.192.1-24.dc.4.14, 312.192.1-24.dc.4.15, 312.192.1-24.dc.4.16
Cyclic 24-isogeny field degree: $2$
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.d

Models

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

$ 0 $ $=$ $ 2 x^{2} - 4 x y - w^{2} $
$=$ $3 x^{2} - 2 y^{2} + 2 y z + z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 5 x^{4} + 2 x^{3} y + 2 x^{2} y^{2} + 4 x^{2} z^{2} - 4 x y z^{2} - 4 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

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}{5^2}\cdot\frac{16544501953020885316042752xz^{23}+49198725590863595615305728xz^{21}w^{2}+55962764865352574947123200xz^{19}w^{4}+35680210101936968091955200xz^{17}w^{6}+9811379907480975837696000xz^{15}w^{8}-1276311709703998252800000xz^{13}w^{10}-2046649537187624092800000xz^{11}w^{12}-411584235281056200000000xz^{9}w^{14}+118848602930372370000000xz^{7}w^{16}+33606853587277725000000xz^{5}w^{18}-4511369513897812500000xz^{3}w^{20}-240497542818281250000xzw^{22}-75334033203057637404311552y^{2}z^{22}-111021313517555139948380160y^{2}z^{20}w^{2}-89889528068629288390656000y^{2}z^{18}w^{4}-32176028232739482187776000y^{2}z^{16}w^{6}-1124287034891912755200000y^{2}z^{14}w^{8}+4950625396355846323200000y^{2}z^{12}w^{10}+1469970377553194176000000y^{2}z^{10}w^{12}-104097831532264800000000y^{2}z^{8}w^{14}-131094785181936600000000y^{2}z^{6}w^{16}+5673163813647500000000y^{2}z^{4}w^{18}+1640463001038750000000y^{2}z^{2}w^{20}+22963486865625000000y^{2}w^{22}-48155810546942362595688448yz^{23}-79240458943434417393598464yz^{21}w^{2}-71719182866343359821824000yz^{19}w^{4}-33033337642882917457305600yz^{17}w^{6}-6000480489395388146688000yz^{15}w^{8}+2717466874834271961600000yz^{13}w^{10}+1634495116253105926400000yz^{11}w^{12}+246796842928269360000000yz^{9}w^{14}-108073542559531860000000yz^{7}w^{16}-21627522663292550000000yz^{5}w^{18}+3371640230105625000000yz^{3}w^{20}+185913731719687500000yzw^{22}-7533403320346181297844224z^{24}-25430090311482872683831296z^{22}w^{2}-34443316412574289914378240z^{20}w^{4}-26302605961594586921011200z^{18}w^{6}-10688546227416566902560000z^{16}w^{8}-960181923627640540800000z^{14}w^{10}+1262558009556580501600000z^{12}w^{12}+575985677790166764000000z^{10}w^{14}+8322502076655198750000z^{8}w^{16}-49535471180788562500000z^{6}w^{18}+35893037698828125000z^{4}w^{20}+850234145700234375000z^{2}w^{22}+5797211487158203125w^{24}}{w^{2}(735041765376xz^{21}+8304151394304xz^{19}w^{2}-397851536320392330240xz^{17}w^{4}-1023193417052356454400xz^{15}w^{6}-878467410322888320000xz^{13}w^{8}-444190322175804000000xz^{11}w^{10}-148826079097275600000xz^{9}w^{12}-32735457200034000000xz^{7}w^{14}-4457695014990000000xz^{5}w^{16}-328228935328125000xz^{3}w^{18}-8857093710937500xzw^{20}-4559079499776y^{2}z^{20}-84808193771520y^{2}z^{18}w^{2}+1811583770836093068800y^{2}z^{16}w^{4}+1941647012010896640000y^{2}z^{14}w^{6}+1126008110701816800000y^{2}z^{12}w^{8}+427304767256781600000y^{2}z^{10}w^{10}+106910641916343000000y^{2}z^{8}w^{12}+17225822987730000000y^{2}z^{6}w^{14}+1669032675421875000y^{2}z^{4}w^{16}+80429601796875000y^{2}z^{2}w^{18}+1087562636718750y^{2}w^{20}+4559079499776yz^{21}+85175714654208yz^{19}w^{2}+1158020723518187153920yz^{17}w^{4}+1440091250922967603200yz^{15}w^{6}+982734935255945760000yz^{13}w^{8}+439240776368316000000yz^{11}w^{10}+133350372624493800000yz^{9}w^{12}+27564329958822000000yz^{7}w^{14}+3693049438798125000yz^{5}w^{16}+278434301765625000yz^{3}w^{18}+8052388769531250yzw^{20}+2279539749888z^{22}+42036576003072z^{20}w^{2}+181158796955630869760z^{18}w^{4}+538716322843936473600z^{16}w^{6}+586234882888504080000z^{14}w^{8}+358981754303281200000z^{12}w^{10}+143447114873092500000z^{10}w^{12}+38656172322216000000z^{8}w^{14}+6801511815599062500z^{6}w^{16}+710245926351562500z^{4}w^{18}+34801214150390625z^{2}w^{20}+306222714843750w^{22})}$

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.c.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.3 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.in.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.ff.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fg.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fh.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fi.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fv.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fw.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fx.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fy.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.bg.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.o.2 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.el.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.en.1 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.xx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xy.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xz.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ya.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.yn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.yo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.yp.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.yq.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.xx.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.xy.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.xz.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ya.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.yn.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.yo.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.yp.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.yq.4 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.xx.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.xy.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.xz.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ya.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.yn.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.yo.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.yp.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.yq.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.xx.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.xy.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.xz.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ya.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.yn.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.yo.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.yp.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.yq.2 $312$ $2$ $2$ $5$ $?$ not computed