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Properties

Label 24.96.1.dn.1
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: 24.96.1.1632

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&6\\4&17\end{bmatrix}$, $\begin{bmatrix}7&6\\20&5\end{bmatrix}$, $\begin{bmatrix}13&21\\0&23\end{bmatrix}$, $\begin{bmatrix}17&9\\16&1\end{bmatrix}$, $\begin{bmatrix}17&18\\4&7\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035917
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dn.1.1, 24.192.1-24.dn.1.2, 24.192.1-24.dn.1.3, 24.192.1-24.dn.1.4, 24.192.1-24.dn.1.5, 24.192.1-24.dn.1.6, 24.192.1-24.dn.1.7, 24.192.1-24.dn.1.8, 24.192.1-24.dn.1.9, 24.192.1-24.dn.1.10, 24.192.1-24.dn.1.11, 24.192.1-24.dn.1.12, 24.192.1-24.dn.1.13, 24.192.1-24.dn.1.14, 24.192.1-24.dn.1.15, 24.192.1-24.dn.1.16, 120.192.1-24.dn.1.1, 120.192.1-24.dn.1.2, 120.192.1-24.dn.1.3, 120.192.1-24.dn.1.4, 120.192.1-24.dn.1.5, 120.192.1-24.dn.1.6, 120.192.1-24.dn.1.7, 120.192.1-24.dn.1.8, 120.192.1-24.dn.1.9, 120.192.1-24.dn.1.10, 120.192.1-24.dn.1.11, 120.192.1-24.dn.1.12, 120.192.1-24.dn.1.13, 120.192.1-24.dn.1.14, 120.192.1-24.dn.1.15, 120.192.1-24.dn.1.16, 168.192.1-24.dn.1.1, 168.192.1-24.dn.1.2, 168.192.1-24.dn.1.3, 168.192.1-24.dn.1.4, 168.192.1-24.dn.1.5, 168.192.1-24.dn.1.6, 168.192.1-24.dn.1.7, 168.192.1-24.dn.1.8, 168.192.1-24.dn.1.9, 168.192.1-24.dn.1.10, 168.192.1-24.dn.1.11, 168.192.1-24.dn.1.12, 168.192.1-24.dn.1.13, 168.192.1-24.dn.1.14, 168.192.1-24.dn.1.15, 168.192.1-24.dn.1.16, 264.192.1-24.dn.1.1, 264.192.1-24.dn.1.2, 264.192.1-24.dn.1.3, 264.192.1-24.dn.1.4, 264.192.1-24.dn.1.5, 264.192.1-24.dn.1.6, 264.192.1-24.dn.1.7, 264.192.1-24.dn.1.8, 264.192.1-24.dn.1.9, 264.192.1-24.dn.1.10, 264.192.1-24.dn.1.11, 264.192.1-24.dn.1.12, 264.192.1-24.dn.1.13, 264.192.1-24.dn.1.14, 264.192.1-24.dn.1.15, 264.192.1-24.dn.1.16, 312.192.1-24.dn.1.1, 312.192.1-24.dn.1.2, 312.192.1-24.dn.1.3, 312.192.1-24.dn.1.4, 312.192.1-24.dn.1.5, 312.192.1-24.dn.1.6, 312.192.1-24.dn.1.7, 312.192.1-24.dn.1.8, 312.192.1-24.dn.1.9, 312.192.1-24.dn.1.10, 312.192.1-24.dn.1.11, 312.192.1-24.dn.1.12, 312.192.1-24.dn.1.13, 312.192.1-24.dn.1.14, 312.192.1-24.dn.1.15, 312.192.1-24.dn.1.16
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
$(-1:0:1)$, $(0:1:0)$

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}{3^3}\cdot\frac{24x^{2}y^{30}-344088x^{2}y^{28}z^{2}+1649960280x^{2}y^{26}z^{4}-3066732489816x^{2}y^{24}z^{6}+2386554559928568x^{2}y^{22}z^{8}-387447961417342200x^{2}y^{20}z^{10}-182123963105956780104x^{2}y^{18}z^{12}+6566088612768062674248x^{2}y^{16}z^{14}-71083208079417375772344x^{2}y^{14}z^{16}-147546990953575410864456x^{2}y^{12}z^{18}+9332156452668093764932104x^{2}y^{10}z^{20}-77092954230943475184127752x^{2}y^{8}z^{22}+241471794178877965363465128x^{2}y^{6}z^{24}-163195650341926018780456872x^{2}y^{4}z^{26}-3342968955945965598553368x^{2}y^{2}z^{28}+6452218951107020797836312x^{2}z^{30}-24xy^{30}z-1055592xy^{28}z^{3}+12904752168xy^{26}z^{5}-38812246823592xy^{24}z^{7}+26994243536690184xy^{22}z^{9}-7686837882988345224xy^{20}z^{11}-202244913361142266680xy^{18}z^{13}+20434250413459252705464xy^{16}z^{15}-500124317341188827001672xy^{14}z^{17}+5976904202899768970147016xy^{12}z^{19}-37627797506347786446354312xy^{10}z^{21}+106629234579568104258654216xy^{8}z^{23}-4908393562725382878949032xy^{6}z^{25}-409340523316581978928946136xy^{4}z^{27}+170612215317245435170788504xy^{2}z^{29}-6452218951107020797836312xz^{31}-y^{32}+16800y^{30}z^{2}-93317832y^{28}z^{4}+191387624256y^{26}z^{6}-104660893409580y^{24}z^{8}+107923033582511328y^{22}z^{10}-49683207835501644888y^{20}z^{12}+519742189388723708160y^{18}z^{14}+32867361223111019681514y^{16}z^{16}-990833336919815151056544y^{14}z^{18}+12119462308069737240297672y^{12}z^{20}-76750370808436687472178624y^{10}z^{22}+240257267014407464904106932y^{8}z^{24}-237846803406994350296517600y^{6}z^{26}-265870928335285008266687784y^{4}z^{28}+176176827428517627710721408y^{2}z^{30}-12984204345290914105535985z^{32}}{z^{2}y^{2}(y^{2}-27z^{2})^{3}(18x^{2}y^{20}+4698x^{2}y^{18}z^{2}-6456024x^{2}y^{16}z^{4}-1203182424x^{2}y^{14}z^{6}+6562233468x^{2}y^{12}z^{8}+4389215860044x^{2}y^{10}z^{10}+41311421583048x^{2}y^{8}z^{12}-1560266283759480x^{2}y^{6}z^{14}-5512459693036158x^{2}y^{4}z^{16}+77826847931777322x^{2}y^{2}z^{18}+63xy^{20}z+180954xy^{18}z^{3}+38677095xy^{16}z^{5}-2164027752xy^{14}z^{7}-437517306306xy^{12}z^{9}-3468704778180xy^{10}z^{11}+355617141057990xy^{8}z^{13}+1647631153710936xy^{6}z^{15}-44519085405963549xy^{4}z^{17}-11118121133111046xy^{2}z^{19}+450283905890997363xz^{21}-y^{22}-2655y^{20}z^{2}-674487y^{18}z^{4}+133942815y^{16}z^{6}+22765869558y^{14}z^{8}+70070495850y^{12}z^{10}-32013588127374y^{10}z^{12}-54568175875650y^{8}z^{14}+6582083109808923y^{6}z^{16}-22321818415775835y^{4}z^{18}-138976514163888075y^{2}z^{20}+450283905890997363z^{22})}$

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