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Properties

Label 16.48.1.p.1
Level $16$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $16$ $\SL_2$-level: $16$ Newform level: $32$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: yes $\quad(D =$ $-16$)

Other labels

Cummins and Pauli (CP) label: 16E1
Rouse and Zureick-Brown (RZB) label: X333
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.94

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}3&12\\0&3\end{bmatrix}$, $\begin{bmatrix}9&8\\12&7\end{bmatrix}$, $\begin{bmatrix}9&9\\4&15\end{bmatrix}$, $\begin{bmatrix}11&11\\12&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.p.1.1, 16.96.1-16.p.1.2, 16.96.1-16.p.1.3, 16.96.1-16.p.1.4, 16.96.1-16.p.1.5, 16.96.1-16.p.1.6, 16.96.1-16.p.1.7, 16.96.1-16.p.1.8, 32.96.1-16.p.1.1, 32.96.1-16.p.1.2, 32.96.1-16.p.1.3, 32.96.1-16.p.1.4, 48.96.1-16.p.1.1, 48.96.1-16.p.1.2, 48.96.1-16.p.1.3, 48.96.1-16.p.1.4, 48.96.1-16.p.1.5, 48.96.1-16.p.1.6, 48.96.1-16.p.1.7, 48.96.1-16.p.1.8, 80.96.1-16.p.1.1, 80.96.1-16.p.1.2, 80.96.1-16.p.1.3, 80.96.1-16.p.1.4, 80.96.1-16.p.1.5, 80.96.1-16.p.1.6, 80.96.1-16.p.1.7, 80.96.1-16.p.1.8, 96.96.1-16.p.1.1, 96.96.1-16.p.1.2, 96.96.1-16.p.1.3, 96.96.1-16.p.1.4, 112.96.1-16.p.1.1, 112.96.1-16.p.1.2, 112.96.1-16.p.1.3, 112.96.1-16.p.1.4, 112.96.1-16.p.1.5, 112.96.1-16.p.1.6, 112.96.1-16.p.1.7, 112.96.1-16.p.1.8, 160.96.1-16.p.1.1, 160.96.1-16.p.1.2, 160.96.1-16.p.1.3, 160.96.1-16.p.1.4, 176.96.1-16.p.1.1, 176.96.1-16.p.1.2, 176.96.1-16.p.1.3, 176.96.1-16.p.1.4, 176.96.1-16.p.1.5, 176.96.1-16.p.1.6, 176.96.1-16.p.1.7, 176.96.1-16.p.1.8, 208.96.1-16.p.1.1, 208.96.1-16.p.1.2, 208.96.1-16.p.1.3, 208.96.1-16.p.1.4, 208.96.1-16.p.1.5, 208.96.1-16.p.1.6, 208.96.1-16.p.1.7, 208.96.1-16.p.1.8, 224.96.1-16.p.1.1, 224.96.1-16.p.1.2, 224.96.1-16.p.1.3, 224.96.1-16.p.1.4, 240.96.1-16.p.1.1, 240.96.1-16.p.1.2, 240.96.1-16.p.1.3, 240.96.1-16.p.1.4, 240.96.1-16.p.1.5, 240.96.1-16.p.1.6, 240.96.1-16.p.1.7, 240.96.1-16.p.1.8, 272.96.1-16.p.1.1, 272.96.1-16.p.1.2, 272.96.1-16.p.1.3, 272.96.1-16.p.1.4, 272.96.1-16.p.1.5, 272.96.1-16.p.1.6, 272.96.1-16.p.1.7, 272.96.1-16.p.1.8, 304.96.1-16.p.1.1, 304.96.1-16.p.1.2, 304.96.1-16.p.1.3, 304.96.1-16.p.1.4, 304.96.1-16.p.1.5, 304.96.1-16.p.1.6, 304.96.1-16.p.1.7, 304.96.1-16.p.1.8
Cyclic 16-isogeny field degree: $4$
Cyclic 16-torsion field degree: $32$
Full 16-torsion field degree: $512$

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 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve CM $j$-invariant $j$-heightWeierstrass model
32.a1 $-16$$287496$ $= 2^{3} \cdot 3^{3} \cdot 11^{3}$$12.569$$(-1:0:1)$, $(0:0:1)$, $(1:0:1)$, $(0:1:0)$

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 -2^3\,\frac{9201060x^{2}y^{22}+1579401648x^{2}y^{21}z+48840055233x^{2}y^{20}z^{2}+566682844800x^{2}y^{19}z^{3}+3471622014100x^{2}y^{18}z^{4}+13620819572736x^{2}y^{17}z^{5}+38250898001646x^{2}y^{16}z^{6}+82327053523200x^{2}y^{15}z^{7}+141975554092368x^{2}y^{14}z^{8}+202174625148096x^{2}y^{13}z^{9}+242683595653173x^{2}y^{12}z^{10}+248990314204800x^{2}y^{11}z^{11}+220253500168260x^{2}y^{10}z^{12}+168657295467280x^{2}y^{9}z^{13}+111851101540350x^{2}y^{8}z^{14}+64389500256000x^{2}y^{7}z^{15}+32615038121940x^{2}y^{6}z^{16}+14906233503216x^{2}y^{5}z^{17}+6184862812107x^{2}y^{4}z^{18}+2205402595200x^{2}y^{3}z^{19}+602991467100x^{2}y^{2}z^{20}+107908657164x^{2}yz^{21}+9420632991x^{2}z^{22}+823284xy^{23}+365225886xy^{22}z+17785603200xy^{21}z^{2}+274107907484xy^{20}z^{3}+2048555741244xy^{19}z^{4}+9368419714102xy^{18}z^{5}+29906773795200xy^{17}z^{6}+72089059169952xy^{16}z^{7}+137969100371408xy^{15}z^{8}+216846279077112xy^{14}z^{9}+286455401395200xy^{13}z^{10}+323247643484372xy^{12}z^{11}+315134207889116xy^{11}z^{12}+267507492310362xy^{10}z^{13}+198872414601600xy^{9}z^{14}+129943980043728xy^{8}z^{15}+74278980538884xy^{7}z^{16}+36254450353302xy^{6}z^{17}+14335621660800xy^{5}z^{18}+4221013878180xy^{4}z^{19}+809321926644xy^{3}z^{20}+75365387361xy^{2}z^{21}+35937y^{24}+67305600y^{23}z+5700571048y^{22}z^{2}+120194922004y^{21}z^{3}+1083879824164y^{20}z^{4}+5528008483200y^{19}z^{5}+18737351413464y^{18}z^{6}+46417739791832y^{17}z^{7}+89237374700658y^{16}z^{8}+138512651366400y^{15}z^{9}+178319545799456y^{14}z^{10}+193962352050820y^{13}z^{11}+180498679351480y^{12}z^{12}+144851597078400y^{11}z^{13}+100572706231368y^{10}z^{14}+60073838525176y^{9}z^{15}+30171666194187y^{8}z^{16}+12147331612800y^{7}z^{17}+3623630971528y^{6}z^{18}+702983615004y^{5}z^{19}+66309549012y^{4}z^{20}+67305600y^{3}z^{21}+9201060y^{2}z^{22}+823284yz^{23}+35937z^{24}}{28x^{2}y^{22}-11361x^{2}y^{20}z^{2}-124372x^{2}y^{18}z^{4}+1446738x^{2}y^{16}z^{6}+10303920x^{2}y^{14}z^{8}-50619477x^{2}y^{12}z^{10}+85768060x^{2}y^{10}z^{12}-75684798x^{2}y^{8}z^{14}+38532844x^{2}y^{6}z^{16}-11337387x^{2}y^{4}z^{18}+1834980x^{2}y^{2}z^{20}-262143x^{2}z^{22}-350xy^{22}z+30884xy^{20}z^{3}+580618xy^{18}z^{5}-578720xy^{16}z^{7}-22381560xy^{14}z^{9}+77550060xy^{12}z^{11}-113649050xy^{10}z^{13}+91752752xy^{8}z^{15}-44171606xy^{6}z^{17}+12845084xy^{4}z^{19}-2097153xy^{2}z^{21}-y^{24}+2520y^{22}z^{2}-22692y^{20}z^{4}-1215576y^{18}z^{6}-3857842y^{16}z^{8}+31910624y^{14}z^{10}-63878328y^{12}z^{12}+62406456y^{10}z^{14}-34415339y^{8}z^{16}+11012344y^{6}z^{18}-1835348y^{4}z^{20}+28y^{2}z^{22}-z^{24}}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.y.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.24.0.j.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.24.1.d.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
16.96.3.cg.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.ch.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.dv.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
16.96.3.dw.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
32.96.3.q.1 $32$ $2$ $2$ $3$ $1$ $1^{2}$
32.96.3.q.2 $32$ $2$ $2$ $3$ $1$ $1^{2}$
32.96.5.o.1 $32$ $2$ $2$ $5$ $0$ $1^{4}$
32.96.5.q.1 $32$ $2$ $2$ $5$ $2$ $1^{4}$
48.96.3.ib.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.ic.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.jz.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.ka.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.144.9.cl.1 $48$ $3$ $3$ $9$ $4$ $1^{8}$
48.192.9.rt.1 $48$ $4$ $4$ $9$ $0$ $1^{8}$
80.96.3.jv.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.jw.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.mb.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.mc.1 $80$ $2$ $2$ $3$ $?$ not computed
80.240.17.bf.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.ed.1 $80$ $6$ $6$ $17$ $?$ not computed
96.96.3.q.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.q.2 $96$ $2$ $2$ $3$ $?$ not computed
96.96.5.p.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.q.1 $96$ $2$ $2$ $5$ $?$ not computed
112.96.3.ht.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.hu.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.jr.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.js.1 $112$ $2$ $2$ $3$ $?$ not computed
160.96.3.q.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.q.2 $160$ $2$ $2$ $3$ $?$ not computed
160.96.5.p.1 $160$ $2$ $2$ $5$ $?$ not computed
160.96.5.q.1 $160$ $2$ $2$ $5$ $?$ not computed
176.96.3.ht.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.hu.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.jr.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.js.1 $176$ $2$ $2$ $3$ $?$ not computed
208.96.3.jv.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.jw.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.mb.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.mc.1 $208$ $2$ $2$ $3$ $?$ not computed
224.96.3.q.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.q.2 $224$ $2$ $2$ $3$ $?$ not computed
224.96.5.p.1 $224$ $2$ $2$ $5$ $?$ not computed
224.96.5.q.1 $224$ $2$ $2$ $5$ $?$ not computed
240.96.3.bbz.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bca.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bhp.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bhq.1 $240$ $2$ $2$ $3$ $?$ not computed
272.96.3.jn.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.jo.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.mb.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.mc.1 $272$ $2$ $2$ $3$ $?$ not computed
304.96.3.ht.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.hu.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.jr.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.js.1 $304$ $2$ $2$ $3$ $?$ not computed