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Properties

Label 24.96.1.e.2
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: 24.96.1.867

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&16\\4&21\end{bmatrix}$, $\begin{bmatrix}7&8\\14&9\end{bmatrix}$, $\begin{bmatrix}9&4\\10&3\end{bmatrix}$, $\begin{bmatrix}9&4\\20&13\end{bmatrix}$, $\begin{bmatrix}9&8\\2&15\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^4\times \GL(2,3)$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.e.2.1, 24.192.1-24.e.2.2, 24.192.1-24.e.2.3, 24.192.1-24.e.2.4, 24.192.1-24.e.2.5, 24.192.1-24.e.2.6, 24.192.1-24.e.2.7, 24.192.1-24.e.2.8, 24.192.1-24.e.2.9, 24.192.1-24.e.2.10, 24.192.1-24.e.2.11, 24.192.1-24.e.2.12, 120.192.1-24.e.2.1, 120.192.1-24.e.2.2, 120.192.1-24.e.2.3, 120.192.1-24.e.2.4, 120.192.1-24.e.2.5, 120.192.1-24.e.2.6, 120.192.1-24.e.2.7, 120.192.1-24.e.2.8, 120.192.1-24.e.2.9, 120.192.1-24.e.2.10, 120.192.1-24.e.2.11, 120.192.1-24.e.2.12, 168.192.1-24.e.2.1, 168.192.1-24.e.2.2, 168.192.1-24.e.2.3, 168.192.1-24.e.2.4, 168.192.1-24.e.2.5, 168.192.1-24.e.2.6, 168.192.1-24.e.2.7, 168.192.1-24.e.2.8, 168.192.1-24.e.2.9, 168.192.1-24.e.2.10, 168.192.1-24.e.2.11, 168.192.1-24.e.2.12, 264.192.1-24.e.2.1, 264.192.1-24.e.2.2, 264.192.1-24.e.2.3, 264.192.1-24.e.2.4, 264.192.1-24.e.2.5, 264.192.1-24.e.2.6, 264.192.1-24.e.2.7, 264.192.1-24.e.2.8, 264.192.1-24.e.2.9, 264.192.1-24.e.2.10, 264.192.1-24.e.2.11, 264.192.1-24.e.2.12, 312.192.1-24.e.2.1, 312.192.1-24.e.2.2, 312.192.1-24.e.2.3, 312.192.1-24.e.2.4, 312.192.1-24.e.2.5, 312.192.1-24.e.2.6, 312.192.1-24.e.2.7, 312.192.1-24.e.2.8, 312.192.1-24.e.2.9, 312.192.1-24.e.2.10, 312.192.1-24.e.2.11, 312.192.1-24.e.2.12
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 z + y w + 2 z^{2} - 3 z w + w^{2} $
$=$ $6 x^{2} - 2 y^{2} - 2 y z + 2 y w - z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 5 x^{4} - 16 x^{3} z - 6 x^{2} y^{2} + 30 x^{2} z^{2} - 12 x y^{2} z - 16 x z^{3} - 6 y^{2} z^{2} + 5 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:

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

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{1}{3^4}\cdot\frac{4096y^{24}-98304y^{23}w+835584y^{22}w^{2}-3571712y^{21}w^{3}+8577024y^{20}w^{4}-17203200y^{19}w^{5}-20529152y^{18}w^{6}-451510272y^{17}w^{7}-5319954432y^{16}w^{8}-65422032896y^{15}w^{9}-808687730688y^{14}w^{10}-10072250056704y^{13}w^{11}-126301762076672y^{12}w^{12}-1593586936381440y^{11}w^{13}-20219782653247488y^{10}w^{14}-257859760854204416y^{9}w^{15}-3303634931159666688y^{8}w^{16}-42502980763082391552y^{7}w^{17}-548913328373830860800y^{6}w^{18}-7113804399755351457792y^{5}w^{19}-92488311599335053533184y^{4}w^{20}-1205996031159191190536192y^{3}w^{21}-15768077533281757367058432y^{2}w^{22}-206678845552233198190166016yw^{23}-68608192095z^{24}+3291310867464z^{23}w-79338629005884z^{22}w^{2}+1281797112632184z^{21}w^{3}-15619404556783710z^{20}w^{4}+153167025277661400z^{19}w^{5}-1259325243607536972z^{18}w^{6}+8930347708827781800z^{17}w^{7}-55759486485662237745z^{16}w^{8}+311373298925877771216z^{15}w^{9}-1574117170966919261304z^{14}w^{10}+7273665964365151121712z^{13}w^{11}-30955110506487881298468z^{12}w^{12}+122057208149650380075312z^{11}w^{13}-447926827707739772169336z^{10}w^{14}+1534682652546853970033616z^{9}w^{15}-4917037797202568903361585z^{8}w^{16}+14729000086069227679074984z^{7}w^{17}-41135703132378498733757772z^{6}w^{18}+106364741928841156927058136z^{5}w^{19}-250806808484627062587773022z^{4}w^{20}+521198511552245458074243960z^{3}w^{21}-868278741384721373428744764z^{2}w^{22}+713693096404954876494889992zw^{23}-192030880068045097793085023w^{24}}{256y^{16}w^{8}+2048y^{15}w^{9}+27648y^{14}w^{10}+342016y^{13}w^{11}+4279808y^{12}w^{12}+53803008y^{11}w^{13}+679889920y^{10}w^{14}+8634583040y^{9}w^{15}+110176194816y^{8}w^{16}+1411993518080y^{7}w^{17}+18168972206080y^{6}w^{18}+234661599756288y^{5}w^{19}+3041160069324800y^{4}w^{20}+39537108255588352y^{3}w^{21}+515504778088882176y^{2}w^{22}+6739486087935672320yw^{23}+625z^{24}-35500z^{23}w+996650z^{22}w^{2}-18466520z^{21}w^{3}+254363391z^{20}w^{4}-2781407184z^{19}w^{5}+25175663650z^{18}w^{6}-194196282820z^{17}w^{7}+1304285476555z^{16}w^{8}-7754504069840z^{15}w^{9}+41349336060596z^{14}w^{10}-199846201090584z^{13}w^{11}+882977803392906z^{12}w^{12}-3591035534680600z^{11}w^{13}+13515891393548980z^{10}w^{14}-47263794478752464z^{9}w^{15}+153917998221424331z^{8}w^{16}-466994103422770628z^{7}w^{17}+1317098940057592866z^{6}w^{18}-3430532828868814800z^{5}w^{19}+8130733708082202367z^{4}w^{20}-16951659144842643160z^{3}w^{21}+28287347787415450922z^{2}w^{22}-23263986337659620012zw^{23}+6260695060482460273w^{24}}$

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
8.48.1.g.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.a.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.b.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.x.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.y.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.w.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.x.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.o.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.p.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.q.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.r.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.17.kt.1 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.dp.2 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
120.192.5.bk.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bm.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bn.2 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bk.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bl.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bm.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bn.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bk.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bl.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bm.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bn.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bk.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bl.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bm.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bn.2 $312$ $2$ $2$ $5$ $?$ not computed