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Properties

Label 24.96.1.dj.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: $24$ 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 $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ 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: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.2230

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&15\\0&23\end{bmatrix}$, $\begin{bmatrix}7&21\\4&13\end{bmatrix}$, $\begin{bmatrix}11&12\\16&11\end{bmatrix}$, $\begin{bmatrix}19&21\\12&7\end{bmatrix}$, $\begin{bmatrix}23&21\\16&5\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035916
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dj.1.1, 24.192.1-24.dj.1.2, 24.192.1-24.dj.1.3, 24.192.1-24.dj.1.4, 24.192.1-24.dj.1.5, 24.192.1-24.dj.1.6, 24.192.1-24.dj.1.7, 24.192.1-24.dj.1.8, 24.192.1-24.dj.1.9, 24.192.1-24.dj.1.10, 24.192.1-24.dj.1.11, 24.192.1-24.dj.1.12, 24.192.1-24.dj.1.13, 24.192.1-24.dj.1.14, 24.192.1-24.dj.1.15, 24.192.1-24.dj.1.16, 120.192.1-24.dj.1.1, 120.192.1-24.dj.1.2, 120.192.1-24.dj.1.3, 120.192.1-24.dj.1.4, 120.192.1-24.dj.1.5, 120.192.1-24.dj.1.6, 120.192.1-24.dj.1.7, 120.192.1-24.dj.1.8, 120.192.1-24.dj.1.9, 120.192.1-24.dj.1.10, 120.192.1-24.dj.1.11, 120.192.1-24.dj.1.12, 120.192.1-24.dj.1.13, 120.192.1-24.dj.1.14, 120.192.1-24.dj.1.15, 120.192.1-24.dj.1.16, 168.192.1-24.dj.1.1, 168.192.1-24.dj.1.2, 168.192.1-24.dj.1.3, 168.192.1-24.dj.1.4, 168.192.1-24.dj.1.5, 168.192.1-24.dj.1.6, 168.192.1-24.dj.1.7, 168.192.1-24.dj.1.8, 168.192.1-24.dj.1.9, 168.192.1-24.dj.1.10, 168.192.1-24.dj.1.11, 168.192.1-24.dj.1.12, 168.192.1-24.dj.1.13, 168.192.1-24.dj.1.14, 168.192.1-24.dj.1.15, 168.192.1-24.dj.1.16, 264.192.1-24.dj.1.1, 264.192.1-24.dj.1.2, 264.192.1-24.dj.1.3, 264.192.1-24.dj.1.4, 264.192.1-24.dj.1.5, 264.192.1-24.dj.1.6, 264.192.1-24.dj.1.7, 264.192.1-24.dj.1.8, 264.192.1-24.dj.1.9, 264.192.1-24.dj.1.10, 264.192.1-24.dj.1.11, 264.192.1-24.dj.1.12, 264.192.1-24.dj.1.13, 264.192.1-24.dj.1.14, 264.192.1-24.dj.1.15, 264.192.1-24.dj.1.16, 312.192.1-24.dj.1.1, 312.192.1-24.dj.1.2, 312.192.1-24.dj.1.3, 312.192.1-24.dj.1.4, 312.192.1-24.dj.1.5, 312.192.1-24.dj.1.6, 312.192.1-24.dj.1.7, 312.192.1-24.dj.1.8, 312.192.1-24.dj.1.9, 312.192.1-24.dj.1.10, 312.192.1-24.dj.1.11, 312.192.1-24.dj.1.12, 312.192.1-24.dj.1.13, 312.192.1-24.dj.1.14, 312.192.1-24.dj.1.15, 312.192.1-24.dj.1.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
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 w $
$=$ $6 x^{2} - 12 x y - 12 y^{2} - 2 z^{2} - 2 z w - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{4} + 2 x^{3} z + 12 x^{2} y^{2} + x^{2} z^{2} + 12 x y^{2} z - 6 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{1712349618208694599680xy^{23}-8282958968498342068224xy^{21}w^{2}+13055302227156428390400xy^{19}w^{4}-6228736601073858772992xy^{17}w^{6}-681845520822386688000xy^{15}w^{8}-159489681636731535360xy^{13}w^{10}+116150019266946465792xy^{11}w^{12}-92135814556627860480xy^{9}w^{14}+77375275558440240384xy^{7}w^{16}-67679675339280107040xy^{5}w^{18}+61027805611383998976xy^{3}w^{20}-56327864719915634670xyw^{22}+1253526920850044878848y^{24}-5898775883120866492416y^{22}w^{2}+8704295016948827947008y^{20}w^{4}-3000975661964083986432y^{18}w^{6}-1704245489479042080768y^{16}w^{8}+397113581373995040768y^{14}w^{10}-302032834220554278912y^{12}w^{12}+240007317277507494912y^{10}w^{14}-201923560719442412064y^{8}w^{16}+176955903436163650848y^{6}w^{18}-159795445262888890692y^{4}w^{20}+147649612455284275602y^{2}w^{22}-567272382976z^{24}-683563339776z^{23}w-22499494296576z^{22}w^{2}-22336538788864z^{21}w^{3}-351209659810560z^{20}w^{4}-234575467032576z^{19}w^{5}-2752171492213504z^{18}w^{6}-410401156863744z^{17}w^{7}-12410865467147232z^{16}w^{8}+7931804375839232z^{15}w^{9}-41967249456874368z^{14}w^{10}+61981539821132160z^{13}w^{11}-139544379521668384z^{12}w^{12}+270558858404697600z^{11}w^{13}-459409857360588768z^{10}w^{14}+928200879710583968z^{9}w^{15}-1347782147816923998z^{8}w^{16}+2715087412357987248z^{7}w^{17}-3156515742441001996z^{6}w^{18}+6692174007552893292z^{5}w^{19}-4006918616134318971z^{4}w^{20}+12527437647112998992z^{3}w^{21}+12943641070730312625z^{2}w^{22}+9387977453319297021zw^{23}+512w^{24}}{w^{2}(12745074926629158912xy^{21}+8029217329808670720xy^{19}w^{2}+2086633002679861248xy^{17}w^{4}+235033781082292224xy^{15}w^{6}+39397345988124672xy^{13}w^{8}-17586401138878464xy^{11}w^{10}+15901594419191040xy^{9}w^{12}-14276410895196288xy^{7}w^{14}+13120964720639784xy^{5}w^{16}-12282419050989696xy^{3}w^{18}+11671926361046016xyw^{20}+9330042392576262144y^{22}+7104190437073354752y^{20}w^{2}+1884652469460467712y^{18}w^{4}+353070347931205632y^{16}w^{6}-34479417924403200y^{14}w^{8}+48407458220393472y^{12}w^{10}-41526251611999488y^{10}w^{12}+37375319018765376y^{8}w^{14}-34386793021689048y^{6}w^{16}+32216148382854528y^{4}w^{18}-30635429717625024y^{2}w^{20}+25716973824z^{22}+35130039552z^{21}w+165396390912z^{20}w^{2}-29244851712z^{19}w^{3}+464442218496z^{18}w^{4}-828770683392z^{17}w^{5}+1793175798528z^{16}w^{6}-4106944420992z^{15}w^{7}+7734633215904z^{14}w^{8}-16599440096352z^{13}w^{9}+29488711278784z^{12}w^{10}-60042129515040z^{11}w^{11}+98807380315424z^{10}w^{12}-194587169110400z^{9}w^{13}+286193769698832z^{8}w^{14}-558807775562776z^{7}w^{15}+665645748284045z^{6}w^{16}-1372051813979135z^{5}w^{17}+855118954836512z^{4}w^{18}-2579054603369568z^{3}w^{19}-2675379287759840z^{2}w^{20}-1945321060174336zw^{21})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.48.0.bt.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.4 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.is.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.cv.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.dm.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.dv.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.ea.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fb.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fi.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fl.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fp.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.288.9.x.1 $24$ $3$ $3$ $9$ $2$ $1^{4}\cdot2^{2}$
72.288.9.t.3 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.eu.2 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fk.1 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.zv.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bad.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.baf.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bch.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcj.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcp.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcr.2 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.zv.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zx.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bad.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.baf.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bch.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcj.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcp.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcr.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.zv.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zx.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bad.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.baf.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bch.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcj.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcp.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcr.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.zv.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zx.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bad.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.baf.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bch.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcj.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcp.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcr.1 $312$ $2$ $2$ $5$ $?$ not computed