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Properties

Label 16.24.1.c.1
Level $16$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $16$ $\SL_2$-level: $16$ Newform level: $64$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $2^{2}\cdot4\cdot16$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16A1
Rouse and Zureick-Brown (RZB) label: X157
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.24.1.3

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&4\\8&11\end{bmatrix}$, $\begin{bmatrix}5&15\\12&5\end{bmatrix}$, $\begin{bmatrix}7&0\\0&3\end{bmatrix}$, $\begin{bmatrix}11&4\\8&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.48.1-16.c.1.1, 16.48.1-16.c.1.2, 16.48.1-16.c.1.3, 16.48.1-16.c.1.4, 16.48.1-16.c.1.5, 16.48.1-16.c.1.6, 16.48.1-16.c.1.7, 16.48.1-16.c.1.8, 32.48.1-16.c.1.1, 32.48.1-16.c.1.2, 32.48.1-16.c.1.3, 32.48.1-16.c.1.4, 48.48.1-16.c.1.1, 48.48.1-16.c.1.2, 48.48.1-16.c.1.3, 48.48.1-16.c.1.4, 48.48.1-16.c.1.5, 48.48.1-16.c.1.6, 48.48.1-16.c.1.7, 48.48.1-16.c.1.8, 80.48.1-16.c.1.1, 80.48.1-16.c.1.2, 80.48.1-16.c.1.3, 80.48.1-16.c.1.4, 80.48.1-16.c.1.5, 80.48.1-16.c.1.6, 80.48.1-16.c.1.7, 80.48.1-16.c.1.8, 96.48.1-16.c.1.1, 96.48.1-16.c.1.2, 96.48.1-16.c.1.3, 96.48.1-16.c.1.4, 112.48.1-16.c.1.1, 112.48.1-16.c.1.2, 112.48.1-16.c.1.3, 112.48.1-16.c.1.4, 112.48.1-16.c.1.5, 112.48.1-16.c.1.6, 112.48.1-16.c.1.7, 112.48.1-16.c.1.8, 160.48.1-16.c.1.1, 160.48.1-16.c.1.2, 160.48.1-16.c.1.3, 160.48.1-16.c.1.4, 176.48.1-16.c.1.1, 176.48.1-16.c.1.2, 176.48.1-16.c.1.3, 176.48.1-16.c.1.4, 176.48.1-16.c.1.5, 176.48.1-16.c.1.6, 176.48.1-16.c.1.7, 176.48.1-16.c.1.8, 208.48.1-16.c.1.1, 208.48.1-16.c.1.2, 208.48.1-16.c.1.3, 208.48.1-16.c.1.4, 208.48.1-16.c.1.5, 208.48.1-16.c.1.6, 208.48.1-16.c.1.7, 208.48.1-16.c.1.8, 224.48.1-16.c.1.1, 224.48.1-16.c.1.2, 224.48.1-16.c.1.3, 224.48.1-16.c.1.4, 240.48.1-16.c.1.1, 240.48.1-16.c.1.2, 240.48.1-16.c.1.3, 240.48.1-16.c.1.4, 240.48.1-16.c.1.5, 240.48.1-16.c.1.6, 240.48.1-16.c.1.7, 240.48.1-16.c.1.8, 272.48.1-16.c.1.1, 272.48.1-16.c.1.2, 272.48.1-16.c.1.3, 272.48.1-16.c.1.4, 272.48.1-16.c.1.5, 272.48.1-16.c.1.6, 272.48.1-16.c.1.7, 272.48.1-16.c.1.8, 304.48.1-16.c.1.1, 304.48.1-16.c.1.2, 304.48.1-16.c.1.3, 304.48.1-16.c.1.4, 304.48.1-16.c.1.5, 304.48.1-16.c.1.6, 304.48.1-16.c.1.7, 304.48.1-16.c.1.8
Cyclic 16-isogeny field degree: $4$
Cyclic 16-torsion field degree: $32$
Full 16-torsion field degree: $1024$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^4\,\frac{678x^{2}y^{4}z^{2}+4095x^{2}z^{6}+44xy^{6}z+4053xy^{2}z^{5}+y^{8}+4140y^{4}z^{4}+4096z^{8}}{zy^{2}(2x^{2}y^{2}z-xy^{4}+xz^{4}-y^{2}z^{3})}$

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
8.12.0.o.1 $8$ $2$ $2$ $0$ $0$ 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
16.48.1.a.2 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.c.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.m.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.o.1 $16$ $2$ $2$ $1$ $0$ dimension zero
32.48.3.b.1 $32$ $2$ $2$ $3$ $1$ $1^{2}$
32.48.3.b.2 $32$ $2$ $2$ $3$ $1$ $1^{2}$
32.48.3.f.1 $32$ $2$ $2$ $3$ $0$ $2$
32.48.3.f.2 $32$ $2$ $2$ $3$ $0$ $2$
48.48.1.y.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.z.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bc.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bd.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.72.5.c.1 $48$ $3$ $3$ $5$ $0$ $1^{4}$
48.96.5.mj.1 $48$ $4$ $4$ $5$ $1$ $1^{4}$
80.48.1.y.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.48.1.z.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.48.1.bc.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.48.1.bd.1 $80$ $2$ $2$ $1$ $?$ dimension zero
80.120.9.c.1 $80$ $5$ $5$ $9$ $?$ not computed
80.144.9.k.1 $80$ $6$ $6$ $9$ $?$ not computed
80.240.17.co.1 $80$ $10$ $10$ $17$ $?$ not computed
96.48.3.b.1 $96$ $2$ $2$ $3$ $?$ not computed
96.48.3.b.2 $96$ $2$ $2$ $3$ $?$ not computed
96.48.3.f.1 $96$ $2$ $2$ $3$ $?$ not computed
96.48.3.f.2 $96$ $2$ $2$ $3$ $?$ not computed
112.48.1.y.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.48.1.z.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.48.1.bc.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.48.1.bd.1 $112$ $2$ $2$ $1$ $?$ dimension zero
112.192.13.i.1 $112$ $8$ $8$ $13$ $?$ not computed
160.48.3.b.1 $160$ $2$ $2$ $3$ $?$ not computed
160.48.3.b.2 $160$ $2$ $2$ $3$ $?$ not computed
160.48.3.f.1 $160$ $2$ $2$ $3$ $?$ not computed
160.48.3.f.2 $160$ $2$ $2$ $3$ $?$ not computed
176.48.1.y.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.48.1.z.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.48.1.bc.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.48.1.bd.1 $176$ $2$ $2$ $1$ $?$ dimension zero
176.288.21.i.1 $176$ $12$ $12$ $21$ $?$ not computed
208.48.1.y.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.48.1.z.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.48.1.bc.1 $208$ $2$ $2$ $1$ $?$ dimension zero
208.48.1.bd.1 $208$ $2$ $2$ $1$ $?$ dimension zero
224.48.3.b.1 $224$ $2$ $2$ $3$ $?$ not computed
224.48.3.b.2 $224$ $2$ $2$ $3$ $?$ not computed
224.48.3.f.1 $224$ $2$ $2$ $3$ $?$ not computed
224.48.3.f.2 $224$ $2$ $2$ $3$ $?$ not computed
240.48.1.y.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.z.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.bc.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.bd.1 $240$ $2$ $2$ $1$ $?$ dimension zero
272.48.1.y.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.48.1.z.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.48.1.bc.1 $272$ $2$ $2$ $1$ $?$ dimension zero
272.48.1.bd.1 $272$ $2$ $2$ $1$ $?$ dimension zero
304.48.1.y.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.48.1.z.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.48.1.bc.1 $304$ $2$ $2$ $1$ $?$ dimension zero
304.48.1.bd.1 $304$ $2$ $2$ $1$ $?$ dimension zero