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Properties

Label 24.48.1.in.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.505

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&14\\12&11\end{bmatrix}$, $\begin{bmatrix}7&7\\12&5\end{bmatrix}$, $\begin{bmatrix}13&4\\12&23\end{bmatrix}$, $\begin{bmatrix}17&8\\0&17\end{bmatrix}$, $\begin{bmatrix}17&14\\0&23\end{bmatrix}$, $\begin{bmatrix}19&14\\12&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.in.1.1, 24.96.1-24.in.1.2, 24.96.1-24.in.1.3, 24.96.1-24.in.1.4, 24.96.1-24.in.1.5, 24.96.1-24.in.1.6, 24.96.1-24.in.1.7, 24.96.1-24.in.1.8, 24.96.1-24.in.1.9, 24.96.1-24.in.1.10, 24.96.1-24.in.1.11, 24.96.1-24.in.1.12, 24.96.1-24.in.1.13, 24.96.1-24.in.1.14, 24.96.1-24.in.1.15, 24.96.1-24.in.1.16, 24.96.1-24.in.1.17, 24.96.1-24.in.1.18, 24.96.1-24.in.1.19, 24.96.1-24.in.1.20, 120.96.1-24.in.1.1, 120.96.1-24.in.1.2, 120.96.1-24.in.1.3, 120.96.1-24.in.1.4, 120.96.1-24.in.1.5, 120.96.1-24.in.1.6, 120.96.1-24.in.1.7, 120.96.1-24.in.1.8, 120.96.1-24.in.1.9, 120.96.1-24.in.1.10, 120.96.1-24.in.1.11, 120.96.1-24.in.1.12, 120.96.1-24.in.1.13, 120.96.1-24.in.1.14, 120.96.1-24.in.1.15, 120.96.1-24.in.1.16, 120.96.1-24.in.1.17, 120.96.1-24.in.1.18, 120.96.1-24.in.1.19, 120.96.1-24.in.1.20, 168.96.1-24.in.1.1, 168.96.1-24.in.1.2, 168.96.1-24.in.1.3, 168.96.1-24.in.1.4, 168.96.1-24.in.1.5, 168.96.1-24.in.1.6, 168.96.1-24.in.1.7, 168.96.1-24.in.1.8, 168.96.1-24.in.1.9, 168.96.1-24.in.1.10, 168.96.1-24.in.1.11, 168.96.1-24.in.1.12, 168.96.1-24.in.1.13, 168.96.1-24.in.1.14, 168.96.1-24.in.1.15, 168.96.1-24.in.1.16, 168.96.1-24.in.1.17, 168.96.1-24.in.1.18, 168.96.1-24.in.1.19, 168.96.1-24.in.1.20, 264.96.1-24.in.1.1, 264.96.1-24.in.1.2, 264.96.1-24.in.1.3, 264.96.1-24.in.1.4, 264.96.1-24.in.1.5, 264.96.1-24.in.1.6, 264.96.1-24.in.1.7, 264.96.1-24.in.1.8, 264.96.1-24.in.1.9, 264.96.1-24.in.1.10, 264.96.1-24.in.1.11, 264.96.1-24.in.1.12, 264.96.1-24.in.1.13, 264.96.1-24.in.1.14, 264.96.1-24.in.1.15, 264.96.1-24.in.1.16, 264.96.1-24.in.1.17, 264.96.1-24.in.1.18, 264.96.1-24.in.1.19, 264.96.1-24.in.1.20, 312.96.1-24.in.1.1, 312.96.1-24.in.1.2, 312.96.1-24.in.1.3, 312.96.1-24.in.1.4, 312.96.1-24.in.1.5, 312.96.1-24.in.1.6, 312.96.1-24.in.1.7, 312.96.1-24.in.1.8, 312.96.1-24.in.1.9, 312.96.1-24.in.1.10, 312.96.1-24.in.1.11, 312.96.1-24.in.1.12, 312.96.1-24.in.1.13, 312.96.1-24.in.1.14, 312.96.1-24.in.1.15, 312.96.1-24.in.1.16, 312.96.1-24.in.1.17, 312.96.1-24.in.1.18, 312.96.1-24.in.1.19, 312.96.1-24.in.1.20
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 156x - 560 $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(14:0:1)$, $(-4:0:1)$, $(-10:0:1)$

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2\cdot3}\cdot\frac{4224x^{2}y^{14}-229828661280x^{2}y^{12}z^{2}+1865204343548928x^{2}y^{10}z^{4}-2402362921048888320x^{2}y^{8}z^{6}-22637632510917638553600x^{2}y^{6}z^{8}-87995987119928758811295744x^{2}y^{4}z^{10}-132941322956799955535124234240x^{2}y^{2}z^{12}-71293287761579732079472696885248x^{2}z^{14}-6061008xy^{14}z+6541249271040xy^{12}z^{3}-19469620856086272xy^{10}z^{5}-28662573529711153152xy^{8}z^{7}-471599394479036004040704xy^{6}z^{9}-1502332445745398453832253440xy^{4}z^{11}-2026212922912803901055456772096xy^{2}z^{13}-998106321370310908720019920650240xz^{15}-y^{16}+3111018240y^{14}z^{2}-94743779016960y^{12}z^{4}+56962790755246080y^{10}z^{6}-1006963808999269220352y^{8}z^{8}-5641121544337750073278464y^{6}z^{10}-11477687879443480129904836608y^{4}z^{12}-9938531443580129760884268466176y^{2}z^{14}-2851734517041983082643773651419136z^{16}}{y^{2}(x^{2}y^{12}+1718496x^{2}y^{10}z^{2}+69572307456x^{2}y^{8}z^{4}+355155763940352x^{2}y^{6}z^{6}+394918619013402624x^{2}y^{4}z^{8}+123256172596690944x^{2}y^{2}z^{10}+26623333280885243904x^{2}z^{12}+236xy^{12}z+80485488xy^{10}z^{3}+1596111837696xy^{8}z^{5}+5890187067724800xy^{6}z^{7}+5526386252402835456xy^{4}z^{9}-1047677467071873024xy^{2}z^{11}-266233332808852439040xz^{13}+25660y^{12}z^{2}+2596732992y^{10}z^{4}+22223448126720y^{8}z^{6}+39876252372062208y^{6}z^{8}+15802814665837264896y^{4}z^{10}-10600030843315421184y^{2}z^{12}-1490906663729573658624z^{14})}$

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
$X_0(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
24.12.0.v.1 $24$ $4$ $4$ $0$ $0$ full Jacobian
24.24.0.cb.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.er.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
24.96.1.dc.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dc.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dc.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dc.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.fe.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.ff.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.fy.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.fy.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.fz.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.fz.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gc.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.gd.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.144.5.gq.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
72.144.5.bh.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.cs.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.cv.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.1.ra.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ra.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ra.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ra.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.mi.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.mj.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.mm.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.mm.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.mn.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.mn.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.mq.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.mr.1 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.bpz.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.bkfd.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.qy.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qy.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qy.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qy.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.jy.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.jz.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.kc.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.kc.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.kd.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.kd.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.kg.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.kh.1 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.qy.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qy.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qy.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qy.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.jy.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.jz.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.kc.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.kc.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.kd.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.kd.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.kg.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.kh.1 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.ra.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ra.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ra.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ra.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.mi.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.mj.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.mm.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.mm.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.mn.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.mn.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.mq.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.mr.1 $312$ $2$ $2$ $3$ $?$ not computed