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Properties

Label 24.24.1.d.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: $8$ 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 $4^{2}\cdot8^{2}$ 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: 8B1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.24.1.6

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&16\\16&15\end{bmatrix}$, $\begin{bmatrix}11&10\\20&17\end{bmatrix}$, $\begin{bmatrix}19&2\\16&15\end{bmatrix}$, $\begin{bmatrix}19&8\\16&1\end{bmatrix}$, $\begin{bmatrix}23&4\\20&21\end{bmatrix}$, $\begin{bmatrix}23&10\\0&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.48.1-24.d.1.1, 24.48.1-24.d.1.2, 24.48.1-24.d.1.3, 24.48.1-24.d.1.4, 24.48.1-24.d.1.5, 24.48.1-24.d.1.6, 24.48.1-24.d.1.7, 24.48.1-24.d.1.8, 24.48.1-24.d.1.9, 24.48.1-24.d.1.10, 24.48.1-24.d.1.11, 24.48.1-24.d.1.12, 24.48.1-24.d.1.13, 24.48.1-24.d.1.14, 24.48.1-24.d.1.15, 24.48.1-24.d.1.16, 24.48.1-24.d.1.17, 24.48.1-24.d.1.18, 24.48.1-24.d.1.19, 24.48.1-24.d.1.20, 120.48.1-24.d.1.1, 120.48.1-24.d.1.2, 120.48.1-24.d.1.3, 120.48.1-24.d.1.4, 120.48.1-24.d.1.5, 120.48.1-24.d.1.6, 120.48.1-24.d.1.7, 120.48.1-24.d.1.8, 120.48.1-24.d.1.9, 120.48.1-24.d.1.10, 120.48.1-24.d.1.11, 120.48.1-24.d.1.12, 120.48.1-24.d.1.13, 120.48.1-24.d.1.14, 120.48.1-24.d.1.15, 120.48.1-24.d.1.16, 120.48.1-24.d.1.17, 120.48.1-24.d.1.18, 120.48.1-24.d.1.19, 120.48.1-24.d.1.20, 168.48.1-24.d.1.1, 168.48.1-24.d.1.2, 168.48.1-24.d.1.3, 168.48.1-24.d.1.4, 168.48.1-24.d.1.5, 168.48.1-24.d.1.6, 168.48.1-24.d.1.7, 168.48.1-24.d.1.8, 168.48.1-24.d.1.9, 168.48.1-24.d.1.10, 168.48.1-24.d.1.11, 168.48.1-24.d.1.12, 168.48.1-24.d.1.13, 168.48.1-24.d.1.14, 168.48.1-24.d.1.15, 168.48.1-24.d.1.16, 168.48.1-24.d.1.17, 168.48.1-24.d.1.18, 168.48.1-24.d.1.19, 168.48.1-24.d.1.20, 264.48.1-24.d.1.1, 264.48.1-24.d.1.2, 264.48.1-24.d.1.3, 264.48.1-24.d.1.4, 264.48.1-24.d.1.5, 264.48.1-24.d.1.6, 264.48.1-24.d.1.7, 264.48.1-24.d.1.8, 264.48.1-24.d.1.9, 264.48.1-24.d.1.10, 264.48.1-24.d.1.11, 264.48.1-24.d.1.12, 264.48.1-24.d.1.13, 264.48.1-24.d.1.14, 264.48.1-24.d.1.15, 264.48.1-24.d.1.16, 264.48.1-24.d.1.17, 264.48.1-24.d.1.18, 264.48.1-24.d.1.19, 264.48.1-24.d.1.20, 312.48.1-24.d.1.1, 312.48.1-24.d.1.2, 312.48.1-24.d.1.3, 312.48.1-24.d.1.4, 312.48.1-24.d.1.5, 312.48.1-24.d.1.6, 312.48.1-24.d.1.7, 312.48.1-24.d.1.8, 312.48.1-24.d.1.9, 312.48.1-24.d.1.10, 312.48.1-24.d.1.11, 312.48.1-24.d.1.12, 312.48.1-24.d.1.13, 312.48.1-24.d.1.14, 312.48.1-24.d.1.15, 312.48.1-24.d.1.16, 312.48.1-24.d.1.17, 312.48.1-24.d.1.18, 312.48.1-24.d.1.19, 312.48.1-24.d.1.20
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $3072$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 36x $
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^4}\cdot\frac{3888x^{2}y^{4}z^{2}+36xy^{6}z+5038848xy^{2}z^{5}+y^{8}+2176782336z^{8}}{z^{2}y^{4}x^{2}}$

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
$X_{\pm1}(2,4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
24.12.0.bz.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.12.1.bw.1 $24$ $2$ $2$ $1$ $1$ 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
24.48.1.n.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bb.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bg.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bg.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bh.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bh.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bi.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bi.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bj.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bj.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bs.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bv.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.72.5.h.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
24.96.5.h.1 $24$ $4$ $4$ $5$ $2$ $1^{4}$
120.48.1.bw.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.by.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.ce.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.ce.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.cf.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.cf.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.cg.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.cg.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.ch.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.ch.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dc.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.de.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.120.9.d.1 $120$ $5$ $5$ $9$ $?$ not computed
120.144.9.it.1 $120$ $6$ $6$ $9$ $?$ not computed
120.240.17.dv.1 $120$ $10$ $10$ $17$ $?$ not computed
168.48.1.bw.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.by.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.ce.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.ce.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.cf.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.cf.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.cg.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.cg.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.ch.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.ch.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.dc.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.de.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.192.13.d.1 $168$ $8$ $8$ $13$ $?$ not computed
264.48.1.bw.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.by.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.ce.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.ce.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.cf.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.cf.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.cg.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.cg.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.ch.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.ch.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.dc.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.de.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.288.21.d.1 $264$ $12$ $12$ $21$ $?$ not computed
312.48.1.bw.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.by.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.ce.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.ce.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.cf.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.cf.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.cg.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.cg.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.ch.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.ch.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.dc.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.de.1 $312$ $2$ $2$ $1$ $?$ dimension zero