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Properties

Label 24.24.1.eq.1
Level $24$
Index $24$
Genus $1$
Analytic rank $1$
Cusps $4$
$\Q$-cusps $4$

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Invariants

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

Other labels

Cummins and Pauli (CP) label: 12F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.24.1.23

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&16\\18&11\end{bmatrix}$, $\begin{bmatrix}7&22\\0&11\end{bmatrix}$, $\begin{bmatrix}11&7\\0&19\end{bmatrix}$, $\begin{bmatrix}11&13\\0&1\end{bmatrix}$, $\begin{bmatrix}19&5\\0&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.48.1-24.eq.1.1, 24.48.1-24.eq.1.2, 24.48.1-24.eq.1.3, 24.48.1-24.eq.1.4, 24.48.1-24.eq.1.5, 24.48.1-24.eq.1.6, 24.48.1-24.eq.1.7, 24.48.1-24.eq.1.8, 24.48.1-24.eq.1.9, 24.48.1-24.eq.1.10, 24.48.1-24.eq.1.11, 24.48.1-24.eq.1.12, 24.48.1-24.eq.1.13, 24.48.1-24.eq.1.14, 24.48.1-24.eq.1.15, 24.48.1-24.eq.1.16, 120.48.1-24.eq.1.1, 120.48.1-24.eq.1.2, 120.48.1-24.eq.1.3, 120.48.1-24.eq.1.4, 120.48.1-24.eq.1.5, 120.48.1-24.eq.1.6, 120.48.1-24.eq.1.7, 120.48.1-24.eq.1.8, 120.48.1-24.eq.1.9, 120.48.1-24.eq.1.10, 120.48.1-24.eq.1.11, 120.48.1-24.eq.1.12, 120.48.1-24.eq.1.13, 120.48.1-24.eq.1.14, 120.48.1-24.eq.1.15, 120.48.1-24.eq.1.16, 168.48.1-24.eq.1.1, 168.48.1-24.eq.1.2, 168.48.1-24.eq.1.3, 168.48.1-24.eq.1.4, 168.48.1-24.eq.1.5, 168.48.1-24.eq.1.6, 168.48.1-24.eq.1.7, 168.48.1-24.eq.1.8, 168.48.1-24.eq.1.9, 168.48.1-24.eq.1.10, 168.48.1-24.eq.1.11, 168.48.1-24.eq.1.12, 168.48.1-24.eq.1.13, 168.48.1-24.eq.1.14, 168.48.1-24.eq.1.15, 168.48.1-24.eq.1.16, 264.48.1-24.eq.1.1, 264.48.1-24.eq.1.2, 264.48.1-24.eq.1.3, 264.48.1-24.eq.1.4, 264.48.1-24.eq.1.5, 264.48.1-24.eq.1.6, 264.48.1-24.eq.1.7, 264.48.1-24.eq.1.8, 264.48.1-24.eq.1.9, 264.48.1-24.eq.1.10, 264.48.1-24.eq.1.11, 264.48.1-24.eq.1.12, 264.48.1-24.eq.1.13, 264.48.1-24.eq.1.14, 264.48.1-24.eq.1.15, 264.48.1-24.eq.1.16, 312.48.1-24.eq.1.1, 312.48.1-24.eq.1.2, 312.48.1-24.eq.1.3, 312.48.1-24.eq.1.4, 312.48.1-24.eq.1.5, 312.48.1-24.eq.1.6, 312.48.1-24.eq.1.7, 312.48.1-24.eq.1.8, 312.48.1-24.eq.1.9, 312.48.1-24.eq.1.10, 312.48.1-24.eq.1.11, 312.48.1-24.eq.1.12, 312.48.1-24.eq.1.13, 312.48.1-24.eq.1.14, 312.48.1-24.eq.1.15, 312.48.1-24.eq.1.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $3072$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 876x - 9520 $
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Rational points

This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^6\cdot3^6}\cdot\frac{24x^{2}y^{6}+392688x^{2}y^{4}z^{2}+1715447808x^{2}y^{2}z^{4}+2481592329216x^{2}z^{6}+1176xy^{6}z+13374720xy^{4}z^{3}+58582206720xy^{2}z^{5}+85595676880896xz^{7}+y^{8}+19968y^{6}z^{2}+146437632y^{4}z^{4}+524385073152y^{2}z^{6}+720863480254464z^{8}}{z^{4}y^{2}(48x^{2}+1632xz+y^{2}+13440z^{2})}$

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
$X_0(6)$ $6$ $2$ $2$ $0$ $0$ full Jacobian
24.6.0.i.1 $24$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.48.1.bz.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.ci.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.dv.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.dw.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.ih.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.ii.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.ik.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.il.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.72.3.qp.1 $24$ $3$ $3$ $3$ $2$ $1^{2}$
72.72.3.cm.1 $72$ $3$ $3$ $3$ $?$ not computed
72.72.5.ba.1 $72$ $3$ $3$ $5$ $?$ not computed
72.72.5.be.1 $72$ $3$ $3$ $5$ $?$ not computed
120.48.1.byk.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.byl.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.byn.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.byo.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.byw.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.byx.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.byz.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bza.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.120.9.xk.1 $120$ $5$ $5$ $9$ $?$ not computed
120.144.9.rve.1 $120$ $6$ $6$ $9$ $?$ not computed
120.240.17.gig.1 $120$ $10$ $10$ $17$ $?$ not computed
168.48.1.byi.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.byj.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.byl.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bym.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.byu.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.byv.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.byx.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.byy.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.192.13.pc.1 $168$ $8$ $8$ $13$ $?$ not computed
264.48.1.byi.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.byj.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.byl.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bym.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.byu.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.byv.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.byx.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.byy.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.288.21.mq.1 $264$ $12$ $12$ $21$ $?$ not computed
312.48.1.byk.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.byl.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.byn.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.byo.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.byw.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.byx.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.byz.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bza.1 $312$ $2$ $2$ $1$ $?$ dimension zero