↑

Properties

Label 24.48.1.n.2
Level $24$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $2$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $8$ 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 $2$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.21

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}3&20\\8&7\end{bmatrix}$, $\begin{bmatrix}5&4\\16&19\end{bmatrix}$, $\begin{bmatrix}11&12\\0&11\end{bmatrix}$, $\begin{bmatrix}15&4\\20&3\end{bmatrix}$, $\begin{bmatrix}17&4\\8&15\end{bmatrix}$, $\begin{bmatrix}21&8\\16&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.n.2.1, 24.96.1-24.n.2.2, 24.96.1-24.n.2.3, 24.96.1-24.n.2.4, 24.96.1-24.n.2.5, 24.96.1-24.n.2.6, 24.96.1-24.n.2.7, 24.96.1-24.n.2.8, 24.96.1-24.n.2.9, 24.96.1-24.n.2.10, 24.96.1-24.n.2.11, 24.96.1-24.n.2.12, 24.96.1-24.n.2.13, 24.96.1-24.n.2.14, 24.96.1-24.n.2.15, 24.96.1-24.n.2.16, 24.96.1-24.n.2.17, 24.96.1-24.n.2.18, 24.96.1-24.n.2.19, 24.96.1-24.n.2.20, 24.96.1-24.n.2.21, 24.96.1-24.n.2.22, 24.96.1-24.n.2.23, 24.96.1-24.n.2.24, 120.96.1-24.n.2.1, 120.96.1-24.n.2.2, 120.96.1-24.n.2.3, 120.96.1-24.n.2.4, 120.96.1-24.n.2.5, 120.96.1-24.n.2.6, 120.96.1-24.n.2.7, 120.96.1-24.n.2.8, 120.96.1-24.n.2.9, 120.96.1-24.n.2.10, 120.96.1-24.n.2.11, 120.96.1-24.n.2.12, 120.96.1-24.n.2.13, 120.96.1-24.n.2.14, 120.96.1-24.n.2.15, 120.96.1-24.n.2.16, 120.96.1-24.n.2.17, 120.96.1-24.n.2.18, 120.96.1-24.n.2.19, 120.96.1-24.n.2.20, 120.96.1-24.n.2.21, 120.96.1-24.n.2.22, 120.96.1-24.n.2.23, 120.96.1-24.n.2.24, 168.96.1-24.n.2.1, 168.96.1-24.n.2.2, 168.96.1-24.n.2.3, 168.96.1-24.n.2.4, 168.96.1-24.n.2.5, 168.96.1-24.n.2.6, 168.96.1-24.n.2.7, 168.96.1-24.n.2.8, 168.96.1-24.n.2.9, 168.96.1-24.n.2.10, 168.96.1-24.n.2.11, 168.96.1-24.n.2.12, 168.96.1-24.n.2.13, 168.96.1-24.n.2.14, 168.96.1-24.n.2.15, 168.96.1-24.n.2.16, 168.96.1-24.n.2.17, 168.96.1-24.n.2.18, 168.96.1-24.n.2.19, 168.96.1-24.n.2.20, 168.96.1-24.n.2.21, 168.96.1-24.n.2.22, 168.96.1-24.n.2.23, 168.96.1-24.n.2.24, 264.96.1-24.n.2.1, 264.96.1-24.n.2.2, 264.96.1-24.n.2.3, 264.96.1-24.n.2.4, 264.96.1-24.n.2.5, 264.96.1-24.n.2.6, 264.96.1-24.n.2.7, 264.96.1-24.n.2.8, 264.96.1-24.n.2.9, 264.96.1-24.n.2.10, 264.96.1-24.n.2.11, 264.96.1-24.n.2.12, 264.96.1-24.n.2.13, 264.96.1-24.n.2.14, 264.96.1-24.n.2.15, 264.96.1-24.n.2.16, 264.96.1-24.n.2.17, 264.96.1-24.n.2.18, 264.96.1-24.n.2.19, 264.96.1-24.n.2.20, 264.96.1-24.n.2.21, 264.96.1-24.n.2.22, 264.96.1-24.n.2.23, 264.96.1-24.n.2.24, 312.96.1-24.n.2.1, 312.96.1-24.n.2.2, 312.96.1-24.n.2.3, 312.96.1-24.n.2.4, 312.96.1-24.n.2.5, 312.96.1-24.n.2.6, 312.96.1-24.n.2.7, 312.96.1-24.n.2.8, 312.96.1-24.n.2.9, 312.96.1-24.n.2.10, 312.96.1-24.n.2.11, 312.96.1-24.n.2.12, 312.96.1-24.n.2.13, 312.96.1-24.n.2.14, 312.96.1-24.n.2.15, 312.96.1-24.n.2.16, 312.96.1-24.n.2.17, 312.96.1-24.n.2.18, 312.96.1-24.n.2.19, 312.96.1-24.n.2.20, 312.96.1-24.n.2.21, 312.96.1-24.n.2.22, 312.96.1-24.n.2.23, 312.96.1-24.n.2.24
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

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} + 9x $
 Copy content Toggle raw display

Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^4}\cdot\frac{5670x^{2}y^{12}z^{2}-268731999x^{2}y^{8}z^{6}+1190026602045x^{2}y^{4}z^{10}-128505439098855x^{2}z^{14}-72xy^{14}z+10491039xy^{10}z^{5}-122463138276xy^{6}z^{9}+71412831316881xy^{2}z^{13}+y^{16}-224532y^{12}z^{4}+5469590772y^{8}z^{8}-6348272132754y^{4}z^{12}+282429536481z^{16}}{z^{2}y^{8}(x^{2}y^{4}-10935x^{2}z^{4}+1377xy^{2}z^{3}-54y^{4}z^{2}+6561z^{6})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{sp}}(4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.l.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.d.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
24.96.1.a.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.a.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.g.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.g.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.r.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.r.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.w.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.w.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.3.m.1 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.m.2 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.o.1 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.96.3.o.2 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.96.3.r.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.r.2 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.t.1 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.t.2 $24$ $2$ $2$ $3$ $1$ $2$
24.144.9.ct.1 $24$ $3$ $3$ $9$ $2$ $1^{8}$
24.192.9.bj.2 $24$ $4$ $4$ $9$ $3$ $1^{8}$
120.96.1.m.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.m.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.w.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.w.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bv.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bv.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ci.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ci.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.bn.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bn.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bu.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bu.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bv.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bv.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.by.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.by.2 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.ba.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.qd.2 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.m.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.m.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.w.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.w.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bv.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bv.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ci.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ci.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.bf.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bf.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bm.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bm.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bn.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bn.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bq.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bq.2 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.m.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.m.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.w.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.w.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.bv.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.bv.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ci.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ci.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.bf.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bf.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bm.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bm.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bn.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bn.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bq.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bq.2 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.m.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.m.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.w.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.w.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.bv.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.bv.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ci.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ci.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.bn.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bn.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bu.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bu.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bv.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bv.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.by.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.by.2 $312$ $2$ $2$ $3$ $?$ not computed