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Properties

Label 24.384.5-24.f.2.12
Level $24$
Index $384$
Genus $5$
Analytic rank $0$
Cusps $24$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $192$
Index: $384$ $\PSL_2$-index:$192$
Genus: $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps: $24$ (none of which are rational) Cusp widths $4^{12}\cdot12^{12}$ Cusp orbits $2^{8}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12E5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.384.5.548

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&8\\12&11\end{bmatrix}$, $\begin{bmatrix}7&20\\12&23\end{bmatrix}$, $\begin{bmatrix}13&8\\0&13\end{bmatrix}$, $\begin{bmatrix}19&8\\0&1\end{bmatrix}$, $\begin{bmatrix}19&12\\18&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_{12}:C_2^4$
Contains $-I$: no $\quad$ (see 24.192.5.f.2 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $192$

Jacobian

Conductor: $2^{27}\cdot3^{5}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2$
Newforms: 24.2.a.a, 192.2.a.b, 192.2.a.d, 192.2.c.a

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

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

$ 0 $ $=$ $ 4 x^{4} y^{2} - 24 x^{4} y z + 40 x^{4} z^{2} + y^{4} z^{2} - 12 y^{3} z^{3} + 51 y^{2} z^{4} + \cdots + 54 z^{6} $
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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 to other modular curves

$j$-invariant map of degree 192 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{34992yz^{20}t^{3}-288684yz^{19}t^{4}+1014768yz^{18}t^{5}-647352yz^{17}t^{6}-17653464yz^{16}t^{7}+166968702yz^{15}t^{8}-1075356648yz^{14}t^{9}+5640999084yz^{13}t^{10}-23372845164yz^{12}t^{11}+53509014072yz^{11}t^{12}+274155785244yz^{10}t^{13}-5151817467504yz^{9}t^{14}+48002827660380yz^{8}t^{15}-345156454952037yz^{7}t^{16}+2031800688302364yz^{6}t^{17}-9290667583992618yz^{5}t^{18}+22715852659653258yz^{4}t^{19}+133992236471876586yz^{3}t^{20}-2606520264831081666yz^{2}t^{21}+25412361380765573076yzt^{22}-190111896013890179502yt^{23}-729z^{24}+17496z^{22}t^{2}-69984z^{21}t^{3}+96228z^{20}t^{4}+594864z^{19}t^{5}-6266484z^{18}t^{6}+36671616z^{17}t^{7}-163902528z^{16}t^{8}+534817728z^{15}t^{9}-386998398z^{14}t^{10}-13533051024z^{13}t^{11}+159762436398z^{12}t^{12}-1267583607864z^{11}t^{13}+8138985095232z^{10}t^{14}-42708709424952z^{9}t^{15}+162277997095305z^{8}t^{16}-116842267832616z^{7}t^{17}-5504482354771611z^{6}t^{18}+70300889627742936z^{5}t^{19}-598542808632531561z^{4}t^{20}+4076300149537003740z^{3}t^{21}-22440065395907177694z^{2}t^{22}+17496zw^{23}+670680zw^{22}t-16883640zw^{21}t^{2}+103908744zw^{20}t^{3}+902472840zw^{19}t^{4}-23547813912zw^{18}t^{5}+246633781176zw^{17}t^{6}-1698431740776zw^{16}t^{7}+8313982020540zw^{15}t^{8}-26765301954276zw^{14}t^{9}+19997205619320zw^{13}t^{10}+444331350551592zw^{12}t^{11}-3976029965332260zw^{11}t^{12}+22560392319464376zw^{10}t^{13}-102037618275061608zw^{9}t^{14}+393666509834583876zw^{8}t^{15}-1333922881740830478zw^{7}t^{16}+4021434606214923390zw^{6}t^{17}-10804387638729804192zw^{5}t^{18}+25578276720981142140zw^{4}t^{19}-51633215811211184352zw^{3}t^{20}+81581044981792008618zw^{2}t^{21}-65848309849987610004zwt^{22}-153085971894939064890zt^{23}-16768w^{24}-469428w^{23}t+17300226w^{22}t^{2}-188345888w^{21}t^{3}+366292227w^{20}t^{4}+15328557408w^{19}t^{5}-247313804230w^{18}t^{6}+2263118329944w^{17}t^{7}-14965096693491w^{16}t^{8}+75999938729092w^{15}t^{9}-293562732653724w^{14}t^{10}+750887108352006w^{13}t^{11}-74703771448225w^{12}t^{12}-13196945263044378w^{11}t^{13}+99135604701392586w^{10}t^{14}-508393251939093248w^{9}t^{15}+2130666658065626913w^{8}t^{16}-7718854152971664162w^{7}t^{17}+24724049585518683629w^{6}t^{18}-70482994652757803577w^{5}t^{19}+177800167609627879812w^{4}t^{20}-388050340667209019153w^{3}t^{21}+679796451593389602381w^{2}t^{22}-566805622524207221355wt^{23}+339140381470160710904t^{24}}{t^{2}(8748yz^{10}t^{11}-196830yz^{9}t^{12}+2283228yz^{8}t^{13}-16824591yz^{7}t^{14}+67350852yz^{6}t^{15}+201877596yz^{5}t^{16}-6797139138yz^{4}t^{17}+76672386189yz^{3}t^{18}-606504364362yz^{2}t^{19}+3579825985740yzt^{20}-13231117565640yt^{21}+729z^{12}t^{10}-17496z^{11}t^{11}+188082z^{10}t^{12}-971028z^{9}t^{13}-2449440z^{8}t^{14}+107477928z^{7}t^{15}-1295286471z^{6}t^{16}+10536117444z^{5}t^{17}-61866925443z^{4}t^{18}+208171449576z^{3}t^{19}+691514303139z^{2}t^{20}-8748zw^{11}t^{10}+145800zw^{10}t^{11}+227448zw^{9}t^{12}-34108452zw^{8}t^{13}+558360054zw^{7}t^{14}-5530052574zw^{6}t^{15}+39469165164zw^{5}t^{16}-212066869824zw^{4}t^{17}+836777697174zw^{3}t^{18}-2136021336180zw^{2}t^{19}+1962899516970zwt^{20}-14564334189492zt^{21}+w^{22}-46w^{21}t+1025w^{20}t^{2}-14810w^{19}t^{3}+156748w^{18}t^{4}-1301626w^{17}t^{5}+8868527w^{16}t^{6}-51182354w^{15}t^{7}+256261711w^{14}t^{8}-1134126540w^{13}t^{9}+4503824347w^{12}t^{10}-16248336282w^{11}t^{11}+53805189262w^{10}t^{12}-164956439648w^{9}t^{13}+471430799162w^{8}t^{14}-1261438321204w^{7}t^{15}+3161334196534w^{6}t^{16}-7379635021187w^{5}t^{17}+15851285021096w^{4}t^{18}-30692978330203w^{3}t^{19}+50061088909567w^{2}t^{20}-40760989089120wt^{21}+23904836500680t^{22})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.f.2 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 2z+2t$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}y+\frac{1}{3}z+\frac{1}{3}w+\frac{1}{3}t$

Equation of the image curve:

$0$ $=$ $ 4X^{4}Y^{2}-24X^{4}YZ+40X^{4}Z^{2}+Y^{4}Z^{2}-12Y^{3}Z^{3}+51Y^{2}Z^{4}-90YZ^{5}+54Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.192.1-12.b.4.2 $12$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.192.1-12.b.4.18 $24$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.192.1-24.cj.1.8 $24$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.192.1-24.cj.1.9 $24$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.192.1-24.cl.1.8 $24$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.192.1-24.cl.1.9 $24$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.192.3-24.f.1.5 $24$ $2$ $2$ $3$ $0$ $2$
24.192.3-24.f.1.21 $24$ $2$ $2$ $3$ $0$ $2$
24.192.3-24.bt.2.6 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.192.3-24.bt.2.15 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.192.3-24.bv.2.8 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.192.3-24.bv.2.22 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.192.3-24.bw.2.4 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.192.3-24.bw.2.15 $24$ $2$ $2$ $3$ $0$ $1^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.768.13-24.i.4.3 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.768.13-24.j.2.10 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.768.13-24.q.1.10 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.768.13-24.r.3.3 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.768.13-24.s.3.1 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.768.13-24.t.1.9 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.768.13-24.bg.4.1 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.768.13-24.bh.2.9 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.768.17-24.ff.2.7 $24$ $2$ $2$ $17$ $0$ $1^{6}\cdot2\cdot4$
24.768.17-24.gl.1.12 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2\cdot4$
24.768.17-24.ha.2.7 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2\cdot4$
24.768.17-24.hc.1.12 $24$ $2$ $2$ $17$ $1$ $1^{6}\cdot2\cdot4$
24.1152.25-24.h.2.15 $24$ $3$ $3$ $25$ $2$ $1^{10}\cdot2^{5}$