↑

Properties

Label 24.48.2.f.2
Level $24$
Index $48$
Genus $2$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $6$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $48$
Index: $48$ $\PSL_2$-index:$48$
Genus: $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $2^{2}\cdot6^{2}\cdot8\cdot24$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24F2
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.2.15

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&21\\16&11\end{bmatrix}$, $\begin{bmatrix}7&6\\12&23\end{bmatrix}$, $\begin{bmatrix}13&3\\4&1\end{bmatrix}$, $\begin{bmatrix}17&0\\12&13\end{bmatrix}$, $\begin{bmatrix}19&15\\8&19\end{bmatrix}$, $\begin{bmatrix}19&18\\4&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.2-24.f.2.1, 24.96.2-24.f.2.2, 24.96.2-24.f.2.3, 24.96.2-24.f.2.4, 24.96.2-24.f.2.5, 24.96.2-24.f.2.6, 24.96.2-24.f.2.7, 24.96.2-24.f.2.8, 24.96.2-24.f.2.9, 24.96.2-24.f.2.10, 24.96.2-24.f.2.11, 24.96.2-24.f.2.12, 24.96.2-24.f.2.13, 24.96.2-24.f.2.14, 24.96.2-24.f.2.15, 24.96.2-24.f.2.16, 24.96.2-24.f.2.17, 24.96.2-24.f.2.18, 24.96.2-24.f.2.19, 24.96.2-24.f.2.20, 24.96.2-24.f.2.21, 24.96.2-24.f.2.22, 24.96.2-24.f.2.23, 24.96.2-24.f.2.24, 24.96.2-24.f.2.25, 24.96.2-24.f.2.26, 24.96.2-24.f.2.27, 24.96.2-24.f.2.28, 24.96.2-24.f.2.29, 24.96.2-24.f.2.30, 24.96.2-24.f.2.31, 24.96.2-24.f.2.32, 120.96.2-24.f.2.1, 120.96.2-24.f.2.2, 120.96.2-24.f.2.3, 120.96.2-24.f.2.4, 120.96.2-24.f.2.5, 120.96.2-24.f.2.6, 120.96.2-24.f.2.7, 120.96.2-24.f.2.8, 120.96.2-24.f.2.9, 120.96.2-24.f.2.10, 120.96.2-24.f.2.11, 120.96.2-24.f.2.12, 120.96.2-24.f.2.13, 120.96.2-24.f.2.14, 120.96.2-24.f.2.15, 120.96.2-24.f.2.16, 120.96.2-24.f.2.17, 120.96.2-24.f.2.18, 120.96.2-24.f.2.19, 120.96.2-24.f.2.20, 120.96.2-24.f.2.21, 120.96.2-24.f.2.22, 120.96.2-24.f.2.23, 120.96.2-24.f.2.24, 120.96.2-24.f.2.25, 120.96.2-24.f.2.26, 120.96.2-24.f.2.27, 120.96.2-24.f.2.28, 120.96.2-24.f.2.29, 120.96.2-24.f.2.30, 120.96.2-24.f.2.31, 120.96.2-24.f.2.32, 168.96.2-24.f.2.1, 168.96.2-24.f.2.2, 168.96.2-24.f.2.3, 168.96.2-24.f.2.4, 168.96.2-24.f.2.5, 168.96.2-24.f.2.6, 168.96.2-24.f.2.7, 168.96.2-24.f.2.8, 168.96.2-24.f.2.9, 168.96.2-24.f.2.10, 168.96.2-24.f.2.11, 168.96.2-24.f.2.12, 168.96.2-24.f.2.13, 168.96.2-24.f.2.14, 168.96.2-24.f.2.15, 168.96.2-24.f.2.16, 168.96.2-24.f.2.17, 168.96.2-24.f.2.18, 168.96.2-24.f.2.19, 168.96.2-24.f.2.20, 168.96.2-24.f.2.21, 168.96.2-24.f.2.22, 168.96.2-24.f.2.23, 168.96.2-24.f.2.24, 168.96.2-24.f.2.25, 168.96.2-24.f.2.26, 168.96.2-24.f.2.27, 168.96.2-24.f.2.28, 168.96.2-24.f.2.29, 168.96.2-24.f.2.30, 168.96.2-24.f.2.31, 168.96.2-24.f.2.32, 264.96.2-24.f.2.1, 264.96.2-24.f.2.2, 264.96.2-24.f.2.3, 264.96.2-24.f.2.4, 264.96.2-24.f.2.5, 264.96.2-24.f.2.6, 264.96.2-24.f.2.7, 264.96.2-24.f.2.8, 264.96.2-24.f.2.9, 264.96.2-24.f.2.10, 264.96.2-24.f.2.11, 264.96.2-24.f.2.12, 264.96.2-24.f.2.13, 264.96.2-24.f.2.14, 264.96.2-24.f.2.15, 264.96.2-24.f.2.16, 264.96.2-24.f.2.17, 264.96.2-24.f.2.18, 264.96.2-24.f.2.19, 264.96.2-24.f.2.20, 264.96.2-24.f.2.21, 264.96.2-24.f.2.22, 264.96.2-24.f.2.23, 264.96.2-24.f.2.24, 264.96.2-24.f.2.25, 264.96.2-24.f.2.26, 264.96.2-24.f.2.27, 264.96.2-24.f.2.28, 264.96.2-24.f.2.29, 264.96.2-24.f.2.30, 264.96.2-24.f.2.31, 264.96.2-24.f.2.32, 312.96.2-24.f.2.1, 312.96.2-24.f.2.2, 312.96.2-24.f.2.3, 312.96.2-24.f.2.4, 312.96.2-24.f.2.5, 312.96.2-24.f.2.6, 312.96.2-24.f.2.7, 312.96.2-24.f.2.8, 312.96.2-24.f.2.9, 312.96.2-24.f.2.10, 312.96.2-24.f.2.11, 312.96.2-24.f.2.12, 312.96.2-24.f.2.13, 312.96.2-24.f.2.14, 312.96.2-24.f.2.15, 312.96.2-24.f.2.16, 312.96.2-24.f.2.17, 312.96.2-24.f.2.18, 312.96.2-24.f.2.19, 312.96.2-24.f.2.20, 312.96.2-24.f.2.21, 312.96.2-24.f.2.22, 312.96.2-24.f.2.23, 312.96.2-24.f.2.24, 312.96.2-24.f.2.25, 312.96.2-24.f.2.26, 312.96.2-24.f.2.27, 312.96.2-24.f.2.28, 312.96.2-24.f.2.29, 312.96.2-24.f.2.30, 312.96.2-24.f.2.31, 312.96.2-24.f.2.32
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{8}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $2$
Newforms: 48.2.c.a

Models

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

$ 0 $ $=$ $ 2 x^{2} w - 3 x y w + x z w + y^{2} w - y z w - z^{2} w $
$=$ $2 x^{2} y - 3 x y^{2} + x y z + y^{3} - y^{2} z - y z^{2}$
$=$ $2 x^{3} - x^{2} y + x^{2} z - 2 x y^{2} - x z^{2} + y^{3} - y^{2} z - y z^{2}$
$=$ $2 x^{2} z - 3 x y z + x z^{2} + y^{2} z - y z^{2} - z^{3}$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{3} y^{2} + 9 x^{2} y^{3} + 3 x^{2} y z^{2} + 9 x y^{2} z^{2} + x z^{4} + 2 y z^{4} $
 Copy content Toggle raw display

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ 2x^{5} - 5x^{4} + 5x^{2} - 2x $
 Copy content Toggle raw display

Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:0:1)$, $(1/2:1:0:0)$, $(1:1:0:0)$, $(-1:-2:1:0)$, $(-1:0:1:0)$, $(1/2:0:1:0)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{9}y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle yz+w^{2}$
$\displaystyle Y$ $=$ $\displaystyle 6y^{2}z^{3}w+9yz^{2}w^{3}+3zw^{5}$
$\displaystyle Z$ $=$ $\displaystyle -yz$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^2}\cdot\frac{47770641xy^{9}+445929300xy^{7}w^{2}+371529018xy^{5}w^{4}+1734055290xy^{3}w^{6}+713557188xyw^{8}-5603094xz^{9}-110060289xz^{7}w^{2}-1457323002xz^{5}w^{4}-2985973746xz^{3}w^{6}-1606894494xzw^{8}-23882769y^{10}-230938452y^{8}w^{2}-249433182y^{6}w^{4}-833403978y^{4}w^{6}-604709712y^{2}w^{8}+6718464yz^{9}-1960066674yz^{7}w^{2}-1121881833yz^{5}w^{4}+4611268611yz^{3}w^{6}+298241874yzw^{8}+2794986z^{10}+57619917z^{8}w^{2}-1338013647z^{6}w^{4}-1119772689z^{4}w^{6}+1542713693z^{2}w^{8}+32768w^{10}}{w^{2}(21xy^{3}w^{4}-92xyw^{6}+17226xz^{7}-12825xz^{5}w^{2}+3894xz^{3}w^{4}-644xzw^{6}-21y^{4}w^{4}+92y^{2}w^{6}-12960yz^{7}+9918yz^{5}w^{2}-3237yz^{3}w^{4}+658yzw^{6}-8694z^{8}+7749z^{6}w^{2}-3063z^{4}w^{4}+523z^{2}w^{6})}$

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
$X_0(12)$ $12$ $2$ $2$ $0$ $0$ 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
24.96.3.bk.1 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.cn.1 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.ed.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.eg.1 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.fo.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.fq.2 $24$ $2$ $2$ $3$ $1$ $1$
24.96.3.fw.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.fy.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gk.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gk.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gl.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gl.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.go.2 $24$ $2$ $2$ $3$ $1$ $1$
24.96.3.go.4 $24$ $2$ $2$ $3$ $1$ $1$
24.96.3.gp.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gp.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gs.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gs.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gt.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gt.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gw.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gw.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gx.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gx.4 $24$ $2$ $2$ $3$ $0$ $1$
24.144.7.xr.1 $24$ $3$ $3$ $7$ $0$ $1\cdot2^{2}$
72.144.7.f.2 $72$ $3$ $3$ $7$ $?$ not computed
72.144.10.g.2 $72$ $3$ $3$ $10$ $?$ not computed
72.144.10.k.2 $72$ $3$ $3$ $10$ $?$ not computed
120.96.3.qw.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qx.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ra.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rb.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rm.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rn.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rq.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rr.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tg.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tg.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.th.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.th.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tk.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tk.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tl.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tl.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.to.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.to.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tp.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tp.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ts.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ts.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tt.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tt.4 $120$ $2$ $2$ $3$ $?$ not computed
120.240.18.o.1 $120$ $5$ $5$ $18$ $?$ not computed
120.288.19.bff.2 $120$ $6$ $6$ $19$ $?$ not computed
168.96.3.oi.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.oj.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.om.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.on.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.oy.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.oz.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pc.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pd.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qs.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qs.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qt.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qt.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qw.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qw.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qx.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qx.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ra.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ra.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rb.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rb.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.re.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.re.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rf.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rf.2 $168$ $2$ $2$ $3$ $?$ not computed
264.96.3.oi.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.oj.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.om.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.on.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.oy.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.oz.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pc.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pd.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qs.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qs.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qt.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qt.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qw.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qw.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qx.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qx.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ra.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ra.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rb.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rb.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.re.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.re.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rf.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rf.4 $264$ $2$ $2$ $3$ $?$ not computed
312.96.3.qw.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qx.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ra.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rb.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rm.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rn.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rq.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rr.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tg.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tg.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.th.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.th.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tk.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tk.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tl.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tl.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.to.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.to.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tp.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tp.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ts.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ts.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tt.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tt.4 $312$ $2$ $2$ $3$ $?$ not computed