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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $288$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.133

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&14\\4&3\end{bmatrix}$, $\begin{bmatrix}11&2\\16&11\end{bmatrix}$, $\begin{bmatrix}11&8\\16&21\end{bmatrix}$, $\begin{bmatrix}21&4\\4&23\end{bmatrix}$, $\begin{bmatrix}21&16\\8&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.bt.1.1, 24.96.1-24.bt.1.2, 24.96.1-24.bt.1.3, 24.96.1-24.bt.1.4, 24.96.1-24.bt.1.5, 24.96.1-24.bt.1.6, 24.96.1-24.bt.1.7, 24.96.1-24.bt.1.8, 24.96.1-24.bt.1.9, 24.96.1-24.bt.1.10, 24.96.1-24.bt.1.11, 24.96.1-24.bt.1.12, 24.96.1-24.bt.1.13, 24.96.1-24.bt.1.14, 24.96.1-24.bt.1.15, 24.96.1-24.bt.1.16, 120.96.1-24.bt.1.1, 120.96.1-24.bt.1.2, 120.96.1-24.bt.1.3, 120.96.1-24.bt.1.4, 120.96.1-24.bt.1.5, 120.96.1-24.bt.1.6, 120.96.1-24.bt.1.7, 120.96.1-24.bt.1.8, 120.96.1-24.bt.1.9, 120.96.1-24.bt.1.10, 120.96.1-24.bt.1.11, 120.96.1-24.bt.1.12, 120.96.1-24.bt.1.13, 120.96.1-24.bt.1.14, 120.96.1-24.bt.1.15, 120.96.1-24.bt.1.16, 168.96.1-24.bt.1.1, 168.96.1-24.bt.1.2, 168.96.1-24.bt.1.3, 168.96.1-24.bt.1.4, 168.96.1-24.bt.1.5, 168.96.1-24.bt.1.6, 168.96.1-24.bt.1.7, 168.96.1-24.bt.1.8, 168.96.1-24.bt.1.9, 168.96.1-24.bt.1.10, 168.96.1-24.bt.1.11, 168.96.1-24.bt.1.12, 168.96.1-24.bt.1.13, 168.96.1-24.bt.1.14, 168.96.1-24.bt.1.15, 168.96.1-24.bt.1.16, 264.96.1-24.bt.1.1, 264.96.1-24.bt.1.2, 264.96.1-24.bt.1.3, 264.96.1-24.bt.1.4, 264.96.1-24.bt.1.5, 264.96.1-24.bt.1.6, 264.96.1-24.bt.1.7, 264.96.1-24.bt.1.8, 264.96.1-24.bt.1.9, 264.96.1-24.bt.1.10, 264.96.1-24.bt.1.11, 264.96.1-24.bt.1.12, 264.96.1-24.bt.1.13, 264.96.1-24.bt.1.14, 264.96.1-24.bt.1.15, 264.96.1-24.bt.1.16, 312.96.1-24.bt.1.1, 312.96.1-24.bt.1.2, 312.96.1-24.bt.1.3, 312.96.1-24.bt.1.4, 312.96.1-24.bt.1.5, 312.96.1-24.bt.1.6, 312.96.1-24.bt.1.7, 312.96.1-24.bt.1.8, 312.96.1-24.bt.1.9, 312.96.1-24.bt.1.10, 312.96.1-24.bt.1.11, 312.96.1-24.bt.1.12, 312.96.1-24.bt.1.13, 312.96.1-24.bt.1.14, 312.96.1-24.bt.1.15, 312.96.1-24.bt.1.16
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ x y - y^{2} - 2 z^{2} $
$=$ $6 x^{2} - 10 x y - 2 y^{2} - 4 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 6 x^{2} y^{2} - 4 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{6}w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^4}\cdot\frac{(144z^{4}+w^{4})^{3}}{w^{4}z^{8}}$

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
8.24.0.h.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.e.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.c.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.bt.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bt.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bv.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bv.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bx.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bx.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bz.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bz.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.jd.1 $24$ $3$ $3$ $9$ $3$ $1^{8}$
24.192.9.ex.1 $24$ $4$ $4$ $9$ $2$ $1^{8}$
120.96.1.kp.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kp.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kr.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kr.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kt.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kt.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kv.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kv.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.240.17.fd.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.fxl.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.kp.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kp.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kr.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kr.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kt.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kt.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kv.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kv.2 $168$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kp.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kp.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kr.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kr.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kt.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kt.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kv.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kv.2 $264$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kp.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kp.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kr.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kr.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kt.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kt.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kv.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kv.2 $312$ $2$ $2$ $1$ $?$ dimension zero