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Properties

Label 24.96.1.dn.3
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: $144$
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 $1^{2}\cdot2^{5}\cdot4$
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: 24.96.1.1608

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&9\\0&23\end{bmatrix}$, $\begin{bmatrix}11&12\\0&13\end{bmatrix}$, $\begin{bmatrix}13&12\\12&5\end{bmatrix}$, $\begin{bmatrix}13&21\\8&13\end{bmatrix}$, $\begin{bmatrix}13&21\\12&5\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035917
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dn.3.1, 24.192.1-24.dn.3.2, 24.192.1-24.dn.3.3, 24.192.1-24.dn.3.4, 24.192.1-24.dn.3.5, 24.192.1-24.dn.3.6, 24.192.1-24.dn.3.7, 24.192.1-24.dn.3.8, 24.192.1-24.dn.3.9, 24.192.1-24.dn.3.10, 24.192.1-24.dn.3.11, 24.192.1-24.dn.3.12, 24.192.1-24.dn.3.13, 24.192.1-24.dn.3.14, 24.192.1-24.dn.3.15, 24.192.1-24.dn.3.16, 120.192.1-24.dn.3.1, 120.192.1-24.dn.3.2, 120.192.1-24.dn.3.3, 120.192.1-24.dn.3.4, 120.192.1-24.dn.3.5, 120.192.1-24.dn.3.6, 120.192.1-24.dn.3.7, 120.192.1-24.dn.3.8, 120.192.1-24.dn.3.9, 120.192.1-24.dn.3.10, 120.192.1-24.dn.3.11, 120.192.1-24.dn.3.12, 120.192.1-24.dn.3.13, 120.192.1-24.dn.3.14, 120.192.1-24.dn.3.15, 120.192.1-24.dn.3.16, 168.192.1-24.dn.3.1, 168.192.1-24.dn.3.2, 168.192.1-24.dn.3.3, 168.192.1-24.dn.3.4, 168.192.1-24.dn.3.5, 168.192.1-24.dn.3.6, 168.192.1-24.dn.3.7, 168.192.1-24.dn.3.8, 168.192.1-24.dn.3.9, 168.192.1-24.dn.3.10, 168.192.1-24.dn.3.11, 168.192.1-24.dn.3.12, 168.192.1-24.dn.3.13, 168.192.1-24.dn.3.14, 168.192.1-24.dn.3.15, 168.192.1-24.dn.3.16, 264.192.1-24.dn.3.1, 264.192.1-24.dn.3.2, 264.192.1-24.dn.3.3, 264.192.1-24.dn.3.4, 264.192.1-24.dn.3.5, 264.192.1-24.dn.3.6, 264.192.1-24.dn.3.7, 264.192.1-24.dn.3.8, 264.192.1-24.dn.3.9, 264.192.1-24.dn.3.10, 264.192.1-24.dn.3.11, 264.192.1-24.dn.3.12, 264.192.1-24.dn.3.13, 264.192.1-24.dn.3.14, 264.192.1-24.dn.3.15, 264.192.1-24.dn.3.16, 312.192.1-24.dn.3.1, 312.192.1-24.dn.3.2, 312.192.1-24.dn.3.3, 312.192.1-24.dn.3.4, 312.192.1-24.dn.3.5, 312.192.1-24.dn.3.6, 312.192.1-24.dn.3.7, 312.192.1-24.dn.3.8, 312.192.1-24.dn.3.9, 312.192.1-24.dn.3.10, 312.192.1-24.dn.3.11, 312.192.1-24.dn.3.12, 312.192.1-24.dn.3.13, 312.192.1-24.dn.3.14, 312.192.1-24.dn.3.15, 312.192.1-24.dn.3.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{4}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 144.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 6x + 7 $
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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}{3^4}\cdot\frac{24x^{2}y^{30}-52488x^{2}y^{28}z^{2}-104188680x^{2}y^{26}z^{4}+174037243464x^{2}y^{24}z^{6}-7694878790952x^{2}y^{22}z^{8}-42031973758391640x^{2}y^{20}z^{10}+11968123213963454616x^{2}y^{18}z^{12}-1354507675385850668232x^{2}y^{16}z^{14}+77128273107660430799496x^{2}y^{14}z^{16}-2192538737195931165138936x^{2}y^{12}z^{18}+16697782733149017651798504x^{2}y^{10}z^{20}+775329781878936387168131448x^{2}y^{8}z^{22}-25120685992665739849298852472x^{2}y^{6}z^{24}+315578620344753639423646330968x^{2}y^{4}z^{26}-1780428640337427264920550486648x^{2}y^{2}z^{28}+3433683811429574365005347993352x^{2}z^{30}-24xy^{30}z+1102248xy^{28}z^{3}-1581935832xy^{26}z^{5}-642836229912xy^{24}z^{7}+1284686992007784xy^{22}z^{9}-370691087882695704xy^{20}z^{11}+36584339574044814120xy^{18}z^{13}-658409918723874558936xy^{16}z^{15}-133442669740081567371912xy^{14}z^{17}+11545893449282646945088056xy^{12}z^{19}-441135030064199161660387272xy^{10}z^{21}+9331956021806706234328999416xy^{8}z^{23}-109903003563441477626887743432xy^{6}z^{25}+631157248567674496316182524984xy^{4}z^{27}-890214331985964458663610037896xy^{2}z^{29}-3433683811429574365005347993352xz^{31}-y^{32}-2640y^{30}z^{2}+4601448y^{28}z^{4}+13624940016y^{26}z^{6}-19081376114220y^{24}z^{8}+5010678654394608y^{22}z^{10}+100519052172885432y^{20}z^{12}-149580741435855529680y^{18}z^{14}+19495855879014591897354y^{16}z^{16}-1230938762501510089861104y^{14}z^{18}+44811881583424646244745752y^{12}z^{20}-977830175181685094411571504y^{10}z^{22}+12336358613640270140362341492y^{8}z^{24}-75362055856259454989516982960y^{6}z^{26}+18840496552486790754293213256y^{4}z^{28}+2034775649731492551812065213968y^{2}z^{30}-6867367702625591806883205850065z^{32}}{z^{2}y^{8}(y^{2}-27z^{2})(x^{2}y^{18}+1701x^{2}y^{16}z^{2}-139968x^{2}y^{14}z^{4}-1102248x^{2}y^{12}z^{6}+36137988x^{2}y^{10}z^{8}+1894055724x^{2}y^{8}z^{10}+8523250758x^{2}y^{6}z^{12}-523017660150x^{2}y^{4}z^{14}+3671583974253x^{2}y^{2}z^{16}-7625597484987x^{2}z^{18}-16xy^{18}z+4374xy^{16}z^{3}+559872xy^{14}z^{5}+11967264xy^{12}z^{7}+136048896xy^{10}z^{9}-2539756539xy^{8}z^{11}-52689186504xy^{6}z^{13}+899590375458xy^{4}z^{15}-4518872583696xy^{2}z^{17}+7625597484987xz^{19}-80y^{18}z^{2}-56376y^{16}z^{4}-225261y^{14}z^{6}-18994095y^{12}z^{8}-287509581y^{10}z^{10}+11063007297y^{8}z^{12}+186349255209y^{6}z^{14}-4508412230493y^{4}z^{16}+27395665038657y^{2}z^{18}-53379182394909z^{20})}$

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
12.48.0.c.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bt.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iu.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.cx.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dm.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.eh.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.em.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ev.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ez.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fr.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fy.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.t.1 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
72.288.9.x.1 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.ey.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fo.3 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.bal.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ban.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbb.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbd.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcx.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcz.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdn.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdp.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bal.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ban.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbb.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbd.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcx.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcz.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdn.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdp.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bal.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ban.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbb.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbd.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcx.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcz.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdn.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdp.3 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bal.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ban.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbb.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbd.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcx.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcz.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdn.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdp.2 $312$ $2$ $2$ $5$ $?$ not computed