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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $64$
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 $4^{8}\cdot8^{8}$ Cusp orbits $2^{2}\cdot4^{3}$
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: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1096

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&18\\0&23\end{bmatrix}$, $\begin{bmatrix}7&20\\0&7\end{bmatrix}$, $\begin{bmatrix}15&8\\4&11\end{bmatrix}$, $\begin{bmatrix}19&4\\0&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1089047
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.bc.1.1, 24.192.1-24.bc.1.2, 24.192.1-24.bc.1.3, 24.192.1-24.bc.1.4, 24.192.1-24.bc.1.5, 24.192.1-24.bc.1.6, 24.192.1-24.bc.1.7, 24.192.1-24.bc.1.8, 120.192.1-24.bc.1.1, 120.192.1-24.bc.1.2, 120.192.1-24.bc.1.3, 120.192.1-24.bc.1.4, 120.192.1-24.bc.1.5, 120.192.1-24.bc.1.6, 120.192.1-24.bc.1.7, 120.192.1-24.bc.1.8, 168.192.1-24.bc.1.1, 168.192.1-24.bc.1.2, 168.192.1-24.bc.1.3, 168.192.1-24.bc.1.4, 168.192.1-24.bc.1.5, 168.192.1-24.bc.1.6, 168.192.1-24.bc.1.7, 168.192.1-24.bc.1.8, 264.192.1-24.bc.1.1, 264.192.1-24.bc.1.2, 264.192.1-24.bc.1.3, 264.192.1-24.bc.1.4, 264.192.1-24.bc.1.5, 264.192.1-24.bc.1.6, 264.192.1-24.bc.1.7, 264.192.1-24.bc.1.8, 312.192.1-24.bc.1.1, 312.192.1-24.bc.1.2, 312.192.1-24.bc.1.3, 312.192.1-24.bc.1.4, 312.192.1-24.bc.1.5, 312.192.1-24.bc.1.6, 312.192.1-24.bc.1.7, 312.192.1-24.bc.1.8
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

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

$ 0 $ $=$ $ x^{4} - 8 x^{3} z + 3 x^{2} y^{2} + 6 x^{2} z^{2} - 12 x y^{2} z + 4 x z^{3} + 12 y^{2} z^{2} - 2 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:

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

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}{3^2}\cdot\frac{4587699380060160yz^{23}-52758542870691840yz^{22}w+284737324346413056yz^{21}w^{2}-958538005115701248yz^{20}w^{3}+2256924248480452608yz^{19}w^{4}-3951368235204120576yz^{18}w^{5}+5340483738538762752yz^{17}w^{6}-5712558196276106496yz^{16}w^{7}+4921417583427096576yz^{15}w^{8}-3458691678644278272yz^{14}w^{9}+2002019861785406976yz^{13}w^{10}-961408406450209536yz^{12}w^{11}+385033816506341376yz^{11}w^{12}-129009261354976512yz^{10}w^{13}+36194941051236864yz^{9}w^{14}-8487016742154240yz^{8}w^{15}+1654397967461760yz^{7}w^{16}-265515244564032yz^{6}w^{17}+34535463765552yz^{5}w^{18}-3551869540440yz^{4}w^{19}+277775253936yz^{3}w^{20}-15475998672yz^{2}w^{21}+544088070yzw^{22}-8998911yw^{23}-3358429036056576z^{24}+40301148432678912z^{23}w-227090977822126080z^{22}w^{2}+798635663798759424z^{21}w^{3}-1965548571172678656z^{20}w^{4}+3598793124482598912z^{19}w^{5}-5088579900217689600z^{18}w^{6}+5695619724113877504z^{17}w^{7}-5134305938033124864z^{16}w^{8}+3774437566855981056z^{15}w^{9}-2284031601616356864z^{14}w^{10}+1145707794592952832z^{13}w^{11}-478820135632794624z^{12}w^{12}+167252707639024128z^{11}w^{13}-48877310146447872z^{10}w^{14}+11930482203429888z^{9}w^{15}-2420307003773520z^{8}w^{16}+404310510483264z^{7}w^{17}-54773537141520z^{6}w^{18}+5874607126224z^{5}w^{19}-480013966836z^{4}w^{20}+28011429720z^{3}w^{21}-1034182674z^{2}w^{22}+17997822zw^{23}-w^{24}}{w^{4}(63538193664yz^{19}-603612839808yz^{18}w+2654416627488yz^{17}w^{2}-7170413918544yz^{16}w^{3}+13314986572032yz^{15}w^{4}-18022630579776yz^{14}w^{5}+18399879868320yz^{13}w^{6}-14464224547344yz^{12}w^{7}+8861134816368yz^{11}w^{8}-4255646155992yz^{10}w^{9}+1603903565142yz^{9}w^{10}-472845542247yz^{8}w^{11}+108232061904yz^{7}w^{12}-19004878602yz^{6}w^{13}+2515295250yz^{5}w^{14}-244619775yz^{4}w^{15}+16832512yz^{3}w^{16}-771072yz^{2}w^{17}+20992yzw^{18}-256yw^{19}-46513185984z^{20}+465131859840z^{19}w-2154934520928z^{18}w^{2}+6138152682912z^{17}w^{3}-12029555428704z^{16}w^{4}+17200018908288z^{15}w^{5}-18565425527328z^{14}w^{6}+15442579183968z^{13}w^{7}-10017637112412z^{12}w^{8}+5097370726152z^{11}w^{9}-2036206565298z^{10}w^{10}+636276756438z^{9}w^{11}-154310493969z^{8}w^{12}+28681099416z^{7}w^{13}-4011248808z^{6}w^{14}+411210738z^{5}w^{15}-29729793z^{4}w^{16}+1425408z^{3}w^{17}-40448z^{2}w^{18}+512zw^{19})}$

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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
8.48.1.m.2 $8$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.g.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.i.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.p.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.r.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.u.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.w.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
24.288.17.pq.2 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.gg.2 $24$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$