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Properties

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

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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: X482
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.96.1.64

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&2\\0&1\end{bmatrix}$, $\begin{bmatrix}7&12\\0&1\end{bmatrix}$, $\begin{bmatrix}11&0\\0&9\end{bmatrix}$, $\begin{bmatrix}11&2\\0&1\end{bmatrix}$, $\begin{bmatrix}15&10\\0&15\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $C_4^2.C_2^4$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.f.1.1, 16.192.1-16.f.1.2, 16.192.1-16.f.1.3, 16.192.1-16.f.1.4, 16.192.1-16.f.1.5, 16.192.1-16.f.1.6, 16.192.1-16.f.1.7, 16.192.1-16.f.1.8, 16.192.1-16.f.1.9, 16.192.1-16.f.1.10, 16.192.1-16.f.1.11, 16.192.1-16.f.1.12, 32.192.1-16.f.1.1, 32.192.1-16.f.1.2, 32.192.1-16.f.1.3, 32.192.1-16.f.1.4, 32.192.1-16.f.1.5, 32.192.1-16.f.1.6, 32.192.1-16.f.1.7, 32.192.1-16.f.1.8, 48.192.1-16.f.1.1, 48.192.1-16.f.1.2, 48.192.1-16.f.1.3, 48.192.1-16.f.1.4, 48.192.1-16.f.1.5, 48.192.1-16.f.1.6, 48.192.1-16.f.1.7, 48.192.1-16.f.1.8, 48.192.1-16.f.1.9, 48.192.1-16.f.1.10, 48.192.1-16.f.1.11, 48.192.1-16.f.1.12, 80.192.1-16.f.1.1, 80.192.1-16.f.1.2, 80.192.1-16.f.1.3, 80.192.1-16.f.1.4, 80.192.1-16.f.1.5, 80.192.1-16.f.1.6, 80.192.1-16.f.1.7, 80.192.1-16.f.1.8, 80.192.1-16.f.1.9, 80.192.1-16.f.1.10, 80.192.1-16.f.1.11, 80.192.1-16.f.1.12, 96.192.1-16.f.1.1, 96.192.1-16.f.1.2, 96.192.1-16.f.1.3, 96.192.1-16.f.1.4, 96.192.1-16.f.1.5, 96.192.1-16.f.1.6, 96.192.1-16.f.1.7, 96.192.1-16.f.1.8, 112.192.1-16.f.1.1, 112.192.1-16.f.1.2, 112.192.1-16.f.1.3, 112.192.1-16.f.1.4, 112.192.1-16.f.1.5, 112.192.1-16.f.1.6, 112.192.1-16.f.1.7, 112.192.1-16.f.1.8, 112.192.1-16.f.1.9, 112.192.1-16.f.1.10, 112.192.1-16.f.1.11, 112.192.1-16.f.1.12, 160.192.1-16.f.1.1, 160.192.1-16.f.1.2, 160.192.1-16.f.1.3, 160.192.1-16.f.1.4, 160.192.1-16.f.1.5, 160.192.1-16.f.1.6, 160.192.1-16.f.1.7, 160.192.1-16.f.1.8, 176.192.1-16.f.1.1, 176.192.1-16.f.1.2, 176.192.1-16.f.1.3, 176.192.1-16.f.1.4, 176.192.1-16.f.1.5, 176.192.1-16.f.1.6, 176.192.1-16.f.1.7, 176.192.1-16.f.1.8, 176.192.1-16.f.1.9, 176.192.1-16.f.1.10, 176.192.1-16.f.1.11, 176.192.1-16.f.1.12, 208.192.1-16.f.1.1, 208.192.1-16.f.1.2, 208.192.1-16.f.1.3, 208.192.1-16.f.1.4, 208.192.1-16.f.1.5, 208.192.1-16.f.1.6, 208.192.1-16.f.1.7, 208.192.1-16.f.1.8, 208.192.1-16.f.1.9, 208.192.1-16.f.1.10, 208.192.1-16.f.1.11, 208.192.1-16.f.1.12, 224.192.1-16.f.1.1, 224.192.1-16.f.1.2, 224.192.1-16.f.1.3, 224.192.1-16.f.1.4, 224.192.1-16.f.1.5, 224.192.1-16.f.1.6, 224.192.1-16.f.1.7, 224.192.1-16.f.1.8, 240.192.1-16.f.1.1, 240.192.1-16.f.1.2, 240.192.1-16.f.1.3, 240.192.1-16.f.1.4, 240.192.1-16.f.1.5, 240.192.1-16.f.1.6, 240.192.1-16.f.1.7, 240.192.1-16.f.1.8, 240.192.1-16.f.1.9, 240.192.1-16.f.1.10, 240.192.1-16.f.1.11, 240.192.1-16.f.1.12, 272.192.1-16.f.1.1, 272.192.1-16.f.1.2, 272.192.1-16.f.1.3, 272.192.1-16.f.1.4, 272.192.1-16.f.1.5, 272.192.1-16.f.1.6, 272.192.1-16.f.1.7, 272.192.1-16.f.1.8, 272.192.1-16.f.1.9, 272.192.1-16.f.1.10, 272.192.1-16.f.1.11, 272.192.1-16.f.1.12, 304.192.1-16.f.1.1, 304.192.1-16.f.1.2, 304.192.1-16.f.1.3, 304.192.1-16.f.1.4, 304.192.1-16.f.1.5, 304.192.1-16.f.1.6, 304.192.1-16.f.1.7, 304.192.1-16.f.1.8, 304.192.1-16.f.1.9, 304.192.1-16.f.1.10, 304.192.1-16.f.1.11, 304.192.1-16.f.1.12
Cyclic 16-isogeny field degree: $1$
Cyclic 16-torsion field degree: $8$
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 $
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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
$(1:0:1)$, $(-1:0:1)$, $(0:0:1)$, $(0:1:0)$

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{720x^{2}y^{30}+395719188x^{2}y^{28}z^{2}+163957645440x^{2}y^{26}z^{4}+6180704673330x^{2}y^{24}z^{6}+68757623808480x^{2}y^{22}z^{8}+331581463950951x^{2}y^{20}z^{10}+827155650102480x^{2}y^{18}z^{12}+1179864481855275x^{2}y^{16}z^{14}+1016842368486960x^{2}y^{14}z^{16}+537716814073464x^{2}y^{12}z^{18}+169446488656320x^{2}y^{10}z^{20}+31281499154565x^{2}y^{8}z^{22}+4553235595440x^{2}y^{6}z^{24}+188743500741x^{2}y^{4}z^{26}+3019898160x^{2}y^{2}z^{28}+16777215x^{2}z^{30}+179272xy^{30}z+4531118400xy^{28}z^{3}+667384554546xy^{26}z^{5}+16119283078080xy^{24}z^{7}+137508659933832xy^{22}z^{9}+551380638884400xy^{20}z^{11}+1203935362225097xy^{18}z^{13}+1550846607837120xy^{16}z^{15}+1228922323844544xy^{14}z^{17}+611157563850240xy^{12}z^{19}+192642543358905xy^{10}z^{21}+35809977860640xy^{8}z^{23}+1693911792700xy^{6}z^{25}+30198989520xy^{4}z^{27}+184549377xy^{2}z^{29}+y^{32}+16948800y^{30}z^{2}+32028588168y^{28}z^{4}+2120901490080y^{26}z^{6}+32251300021148y^{24}z^{8}+191199903598080y^{22}z^{10}+554413269723614y^{20}z^{12}+883315939052880y^{18}z^{14}+825468334146204y^{16}z^{16}+468607928904000y^{14}z^{18}+162705985921424y^{12}z^{20}+31285229236560y^{10}z^{22}+1505712675646y^{8}z^{24}+27196030080y^{6}z^{26}+167951418y^{4}z^{28}+720y^{2}z^{30}+z^{32}}{y^{2}(x^{2}y^{28}-1304x^{2}y^{26}z^{2}-46040x^{2}y^{24}z^{4}-345488x^{2}y^{22}z^{6}+3274070x^{2}y^{20}z^{8}+38696304x^{2}y^{18}z^{10}-24569439x^{2}y^{16}z^{12}-593608208x^{2}y^{14}z^{14}+1776151887x^{2}y^{12}z^{16}-2348297576x^{2}y^{10}z^{18}+1759247244x^{2}y^{8}z^{20}-801111000x^{2}y^{6}z^{22}+217054999x^{2}y^{4}z^{24}-25165800x^{2}y^{2}z^{26}+4194303x^{2}z^{28}-24xy^{28}z+3300xy^{26}z^{3}+142592xy^{24}z^{5}+1601964xy^{22}z^{7}+891344xy^{20}z^{9}-65957489xy^{18}z^{11}-68035104xy^{16}z^{13}+1072372204xy^{14}z^{15}-2560087816xy^{12}z^{17}+3017278816xy^{10}z^{19}-2084570176xy^{8}z^{21}+880804074xy^{6}z^{23}-226492440xy^{4}z^{25}+41943041xy^{2}z^{27}+244y^{28}z^{2}+3168y^{26}z^{4}-129278y^{24}z^{6}-3142880y^{22}z^{8}-17161248y^{20}z^{10}+52203928y^{18}z^{12}+295715913y^{16}z^{14}-1197368928y^{14}z^{16}+1801129060y^{12}z^{18}-1459605056y^{10}z^{20}+697303945y^{8}z^{22}-201327632y^{6}z^{24}+37748968y^{4}z^{26}-24y^{2}z^{28}+z^{30})}$

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