↑

Properties

Label 16.96.1.f.2
Level $16$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $16$ $\SL_2$-level: $16$ Newform level: $32$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $4$ are rational) Cusp widths $2^{8}\cdot4^{4}\cdot16^{4}$ Cusp orbits $1^{4}\cdot2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16M1
Rouse and Zureick-Brown (RZB) label: X470
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.96.1.35

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&14\\0&3\end{bmatrix}$, $\begin{bmatrix}7&0\\0&9\end{bmatrix}$, $\begin{bmatrix}7&2\\0&13\end{bmatrix}$, $\begin{bmatrix}7&8\\0&3\end{bmatrix}$, $\begin{bmatrix}15&8\\0&11\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $C_4^2.C_2^4$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.f.2.1, 16.192.1-16.f.2.2, 16.192.1-16.f.2.3, 16.192.1-16.f.2.4, 16.192.1-16.f.2.5, 16.192.1-16.f.2.6, 16.192.1-16.f.2.7, 16.192.1-16.f.2.8, 16.192.1-16.f.2.9, 16.192.1-16.f.2.10, 16.192.1-16.f.2.11, 16.192.1-16.f.2.12, 32.192.1-16.f.2.1, 32.192.1-16.f.2.2, 32.192.1-16.f.2.3, 32.192.1-16.f.2.4, 32.192.1-16.f.2.5, 32.192.1-16.f.2.6, 32.192.1-16.f.2.7, 32.192.1-16.f.2.8, 48.192.1-16.f.2.1, 48.192.1-16.f.2.2, 48.192.1-16.f.2.3, 48.192.1-16.f.2.4, 48.192.1-16.f.2.5, 48.192.1-16.f.2.6, 48.192.1-16.f.2.7, 48.192.1-16.f.2.8, 48.192.1-16.f.2.9, 48.192.1-16.f.2.10, 48.192.1-16.f.2.11, 48.192.1-16.f.2.12, 80.192.1-16.f.2.1, 80.192.1-16.f.2.2, 80.192.1-16.f.2.3, 80.192.1-16.f.2.4, 80.192.1-16.f.2.5, 80.192.1-16.f.2.6, 80.192.1-16.f.2.7, 80.192.1-16.f.2.8, 80.192.1-16.f.2.9, 80.192.1-16.f.2.10, 80.192.1-16.f.2.11, 80.192.1-16.f.2.12, 96.192.1-16.f.2.1, 96.192.1-16.f.2.2, 96.192.1-16.f.2.3, 96.192.1-16.f.2.4, 96.192.1-16.f.2.5, 96.192.1-16.f.2.6, 96.192.1-16.f.2.7, 96.192.1-16.f.2.8, 112.192.1-16.f.2.1, 112.192.1-16.f.2.2, 112.192.1-16.f.2.3, 112.192.1-16.f.2.4, 112.192.1-16.f.2.5, 112.192.1-16.f.2.6, 112.192.1-16.f.2.7, 112.192.1-16.f.2.8, 112.192.1-16.f.2.9, 112.192.1-16.f.2.10, 112.192.1-16.f.2.11, 112.192.1-16.f.2.12, 160.192.1-16.f.2.1, 160.192.1-16.f.2.2, 160.192.1-16.f.2.3, 160.192.1-16.f.2.4, 160.192.1-16.f.2.5, 160.192.1-16.f.2.6, 160.192.1-16.f.2.7, 160.192.1-16.f.2.8, 176.192.1-16.f.2.1, 176.192.1-16.f.2.2, 176.192.1-16.f.2.3, 176.192.1-16.f.2.4, 176.192.1-16.f.2.5, 176.192.1-16.f.2.6, 176.192.1-16.f.2.7, 176.192.1-16.f.2.8, 176.192.1-16.f.2.9, 176.192.1-16.f.2.10, 176.192.1-16.f.2.11, 176.192.1-16.f.2.12, 208.192.1-16.f.2.1, 208.192.1-16.f.2.2, 208.192.1-16.f.2.3, 208.192.1-16.f.2.4, 208.192.1-16.f.2.5, 208.192.1-16.f.2.6, 208.192.1-16.f.2.7, 208.192.1-16.f.2.8, 208.192.1-16.f.2.9, 208.192.1-16.f.2.10, 208.192.1-16.f.2.11, 208.192.1-16.f.2.12, 224.192.1-16.f.2.1, 224.192.1-16.f.2.2, 224.192.1-16.f.2.3, 224.192.1-16.f.2.4, 224.192.1-16.f.2.5, 224.192.1-16.f.2.6, 224.192.1-16.f.2.7, 224.192.1-16.f.2.8, 240.192.1-16.f.2.1, 240.192.1-16.f.2.2, 240.192.1-16.f.2.3, 240.192.1-16.f.2.4, 240.192.1-16.f.2.5, 240.192.1-16.f.2.6, 240.192.1-16.f.2.7, 240.192.1-16.f.2.8, 240.192.1-16.f.2.9, 240.192.1-16.f.2.10, 240.192.1-16.f.2.11, 240.192.1-16.f.2.12, 272.192.1-16.f.2.1, 272.192.1-16.f.2.2, 272.192.1-16.f.2.3, 272.192.1-16.f.2.4, 272.192.1-16.f.2.5, 272.192.1-16.f.2.6, 272.192.1-16.f.2.7, 272.192.1-16.f.2.8, 272.192.1-16.f.2.9, 272.192.1-16.f.2.10, 272.192.1-16.f.2.11, 272.192.1-16.f.2.12, 304.192.1-16.f.2.1, 304.192.1-16.f.2.2, 304.192.1-16.f.2.3, 304.192.1-16.f.2.4, 304.192.1-16.f.2.5, 304.192.1-16.f.2.6, 304.192.1-16.f.2.7, 304.192.1-16.f.2.8, 304.192.1-16.f.2.9, 304.192.1-16.f.2.10, 304.192.1-16.f.2.11, 304.192.1-16.f.2.12
Cyclic 16-isogeny field degree: $1$
Cyclic 16-torsion field degree: $4$
Full 16-torsion field degree: $256$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x $
 Copy content Toggle raw display

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps to other modular curves

$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{12x^{2}y^{28}z^{2}+3390x^{2}y^{24}z^{6}-398151x^{2}y^{20}z^{10}+15335685x^{2}y^{16}z^{14}-422576184x^{2}y^{12}z^{18}-2110783365x^{2}y^{8}z^{22}-754974741x^{2}y^{4}z^{26}-16777215x^{2}z^{30}+8xy^{30}z-1026xy^{26}z^{5}+4248xy^{22}z^{9}+1965703xy^{18}z^{13}-71303184xy^{14}z^{17}-2091909225xy^{10}z^{21}-2013265900xy^{6}z^{25}-184549377xy^{2}z^{29}-y^{32}-168y^{28}z^{4}+21892y^{24}z^{8}-780974y^{20}z^{12}+14679876y^{16}z^{16}-920649584y^{12}z^{20}-1409286286y^{8}z^{24}-167772138y^{4}z^{28}-z^{32}}{z^{5}y^{16}(3x^{2}y^{8}z-261x^{2}y^{4}z^{5}-255x^{2}z^{9}-xy^{10}+4xy^{6}z^{4}-769xy^{2}z^{8}+26y^{8}z^{3}-506y^{4}z^{7}-z^{11})}$

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
$X_{\pm1}(2,8)$ $8$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.d.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.v.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.x.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.1.b.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.v.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.x.1 $16$ $2$ $2$ $1$ $0$ 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
16.192.5.m.3 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.t.2 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.ba.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.bb.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.be.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.bf.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.bh.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
$X_{\pm1}(2,16)$ $16$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.a.3 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.a.4 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.b.3 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.b.4 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.e.3 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.e.4 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.9.o.1 $32$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
32.192.9.o.2 $32$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.5.ek.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.eo.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.fg.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fh.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fk.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fl.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fn.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fo.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.288.17.fb.1 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.is.1 $48$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
80.192.5.hq.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.hu.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.jk.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.jl.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.js.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.jt.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.jz.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ka.2 $80$ $2$ $2$ $5$ $?$ not computed
96.192.5.a.3 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.a.4 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.b.3 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.b.4 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.k.3 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.k.4 $96$ $2$ $2$ $5$ $?$ not computed
96.192.9.bw.1 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.bw.2 $96$ $2$ $2$ $9$ $?$ not computed
112.192.5.ek.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.eo.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.fg.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.fh.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.fk.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.fl.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.fn.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.fo.1 $112$ $2$ $2$ $5$ $?$ not computed
160.192.5.e.3 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.e.4 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.f.3 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.f.4 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.o.3 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.o.4 $160$ $2$ $2$ $5$ $?$ not computed
160.192.9.bw.1 $160$ $2$ $2$ $9$ $?$ not computed
160.192.9.bw.2 $160$ $2$ $2$ $9$ $?$ not computed
176.192.5.ek.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.eo.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.fg.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.fh.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.fk.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.fl.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.fn.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.fo.2 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.hq.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.hu.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.jk.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.jl.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.js.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.jt.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.jz.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.ka.1 $208$ $2$ $2$ $5$ $?$ not computed
224.192.5.a.3 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.a.4 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.b.3 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.b.4 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.k.3 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.k.4 $224$ $2$ $2$ $5$ $?$ not computed
224.192.9.bw.1 $224$ $2$ $2$ $9$ $?$ not computed
224.192.9.bw.2 $224$ $2$ $2$ $9$ $?$ not computed
240.192.5.bhc.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bhk.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bni.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bnj.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bnq.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bnr.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bnx.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bny.2 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.hq.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.hu.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.jg.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.jh.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.js.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.jt.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.kd.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.ke.1 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.ek.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.eo.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.fg.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.fh.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.fk.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.fl.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.fn.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.fo.1 $304$ $2$ $2$ $5$ $?$ not computed