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Properties

Label 24.96.1.df.4
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $2$ are rational) Cusp widths $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.2252

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&21\\4&5\end{bmatrix}$, $\begin{bmatrix}11&6\\0&17\end{bmatrix}$, $\begin{bmatrix}13&3\\8&7\end{bmatrix}$, $\begin{bmatrix}17&21\\12&19\end{bmatrix}$, $\begin{bmatrix}19&0\\4&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035865
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.df.4.1, 24.192.1-24.df.4.2, 24.192.1-24.df.4.3, 24.192.1-24.df.4.4, 24.192.1-24.df.4.5, 24.192.1-24.df.4.6, 24.192.1-24.df.4.7, 24.192.1-24.df.4.8, 24.192.1-24.df.4.9, 24.192.1-24.df.4.10, 24.192.1-24.df.4.11, 24.192.1-24.df.4.12, 24.192.1-24.df.4.13, 24.192.1-24.df.4.14, 24.192.1-24.df.4.15, 24.192.1-24.df.4.16, 120.192.1-24.df.4.1, 120.192.1-24.df.4.2, 120.192.1-24.df.4.3, 120.192.1-24.df.4.4, 120.192.1-24.df.4.5, 120.192.1-24.df.4.6, 120.192.1-24.df.4.7, 120.192.1-24.df.4.8, 120.192.1-24.df.4.9, 120.192.1-24.df.4.10, 120.192.1-24.df.4.11, 120.192.1-24.df.4.12, 120.192.1-24.df.4.13, 120.192.1-24.df.4.14, 120.192.1-24.df.4.15, 120.192.1-24.df.4.16, 168.192.1-24.df.4.1, 168.192.1-24.df.4.2, 168.192.1-24.df.4.3, 168.192.1-24.df.4.4, 168.192.1-24.df.4.5, 168.192.1-24.df.4.6, 168.192.1-24.df.4.7, 168.192.1-24.df.4.8, 168.192.1-24.df.4.9, 168.192.1-24.df.4.10, 168.192.1-24.df.4.11, 168.192.1-24.df.4.12, 168.192.1-24.df.4.13, 168.192.1-24.df.4.14, 168.192.1-24.df.4.15, 168.192.1-24.df.4.16, 264.192.1-24.df.4.1, 264.192.1-24.df.4.2, 264.192.1-24.df.4.3, 264.192.1-24.df.4.4, 264.192.1-24.df.4.5, 264.192.1-24.df.4.6, 264.192.1-24.df.4.7, 264.192.1-24.df.4.8, 264.192.1-24.df.4.9, 264.192.1-24.df.4.10, 264.192.1-24.df.4.11, 264.192.1-24.df.4.12, 264.192.1-24.df.4.13, 264.192.1-24.df.4.14, 264.192.1-24.df.4.15, 264.192.1-24.df.4.16, 312.192.1-24.df.4.1, 312.192.1-24.df.4.2, 312.192.1-24.df.4.3, 312.192.1-24.df.4.4, 312.192.1-24.df.4.5, 312.192.1-24.df.4.6, 312.192.1-24.df.4.7, 312.192.1-24.df.4.8, 312.192.1-24.df.4.9, 312.192.1-24.df.4.10, 312.192.1-24.df.4.11, 312.192.1-24.df.4.12, 312.192.1-24.df.4.13, 312.192.1-24.df.4.14, 312.192.1-24.df.4.15, 312.192.1-24.df.4.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} + 3x - 3 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(1:0:1)$

Maps to other modular curves

$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2}\cdot\frac{1456x^{2}y^{30}+7966689408x^{2}y^{28}z^{2}+32076405304320x^{2}y^{26}z^{4}+3158774568329216x^{2}y^{24}z^{6}+74738913625178112x^{2}y^{22}z^{8}+760115779120660480x^{2}y^{20}z^{10}+3640614631552057344x^{2}y^{18}z^{12}+8037686138174963712x^{2}y^{16}z^{14}+19295612392590802944x^{2}y^{14}z^{16}+168702610004777107456x^{2}y^{12}z^{18}+822888455871928467456x^{2}y^{10}z^{20}+988049149294943404032x^{2}y^{8}z^{22}-6244819581607658651648x^{2}y^{6}z^{24}-28540000109292146393088x^{2}y^{4}z^{26}-47709006712645334597632x^{2}y^{2}z^{28}-27262293209257745580032x^{2}z^{30}+737280xy^{30}z+206408540160xy^{28}z^{3}+184624242819072xy^{26}z^{5}+6518171718647808xy^{24}z^{7}+96475355522531328xy^{22}z^{9}+781117757339992064xy^{20}z^{11}+3874043932887220224xy^{18}z^{13}+7766705703430914048xy^{16}z^{15}-50666022130390401024xy^{14}z^{17}-401566426019911434240xy^{12}z^{19}-492426196795829256192xy^{10}z^{21}+5167096418306900361216xy^{8}z^{23}+27475333034143251431424xy^{6}z^{25}+57080428183694195294208xy^{4}z^{27}+47709013256938543120384xy^{2}z^{29}+y^{32}+144948176y^{30}z^{2}+3213491660160y^{28}z^{4}+908309608248320y^{26}z^{6}+29412559729172480y^{24}z^{8}+410736023481417728y^{22}z^{10}+3193478425427312640y^{20}z^{12}+15730316053479161856y^{18}z^{14}+57537753244156035072y^{16}z^{16}+164261323935921995776y^{14}z^{18}+166015275008056623104y^{12}z^{20}-1470571932726388064256y^{10}z^{22}-8546782138996707295232y^{8}z^{24}-19941329691800791678976y^{6}z^{26}-16612318829947608104960y^{4}z^{28}+13631166237508498358272y^{2}z^{30}+27262293490732722290688z^{32}}{y^{2}(y^{2}+8z^{2})(x^{2}y^{26}+344x^{2}y^{24}z^{2}-65024x^{2}y^{22}z^{4}-88956928x^{2}y^{20}z^{6}-21067788288x^{2}y^{18}z^{8}-862905073664x^{2}y^{16}z^{10}-6013476405248x^{2}y^{14}z^{12}+284238972715008x^{2}y^{12}z^{14}+6232344683872256x^{2}y^{10}z^{16}+46862907991916544x^{2}y^{8}z^{18}+138663465827958784x^{2}y^{6}z^{20}+123256009387933696x^{2}y^{4}z^{22}-343597383680x^{2}y^{2}z^{24}+549755813888x^{2}z^{26}-34xy^{26}z-44448xy^{24}z^{3}-19275264xy^{22}z^{5}-3281844224xy^{20}z^{7}-217377710080xy^{18}z^{9}-8888319737856xy^{16}z^{11}-229858814722048xy^{14}z^{13}-3466131603980288xy^{12}z^{15}-28085754664058880xy^{10}z^{17}-109132997814386688xy^{8}z^{19}-154070243663151104xy^{6}z^{21}-103079215104xy^{4}z^{23}+137438953472xy^{2}z^{25}+377y^{26}z^{2}+384168y^{24}z^{4}+133583360y^{22}z^{6}+19616782336y^{20}z^{8}+1131135602688y^{18}z^{10}+33796993875968y^{16}z^{12}+559009973141504y^{14}z^{14}+5000923151597568y^{12}z^{16}+21853313746075648y^{10}z^{18}+31455722940137472y^{8}z^{20}-46220307735773184y^{6}z^{22}-123256730942439424y^{4}z^{24}-893353197568y^{2}z^{26}+1649267441664z^{28})}$

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
24.48.0.bt.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iq.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.dd.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dl.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dp.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.du.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.et.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ez.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fb.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fh.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.o.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.p.3 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.eq.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fg.4 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.zf.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zh.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zp.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbr.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbt.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbz.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcb.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.zf.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zh.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zn.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zp.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbr.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbt.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbz.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcb.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.zf.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zh.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zn.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zp.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbr.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbt.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbz.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcb.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.zf.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zh.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zn.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zp.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbr.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbt.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbz.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcb.4 $312$ $2$ $2$ $5$ $?$ not computed