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Properties

Label 16.48.1.k.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: $8$ 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 $4^{4}\cdot8^{4}$ Cusp orbits $4^{2}$
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 $\quad(D =$ $-8$)

Other labels

Cummins and Pauli (CP) label: 8G1
Rouse and Zureick-Brown (RZB) label: X343
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.198

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&5\\2&7\end{bmatrix}$, $\begin{bmatrix}3&11\\2&11\end{bmatrix}$, $\begin{bmatrix}7&13\\14&15\end{bmatrix}$, $\begin{bmatrix}13&14\\12&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 16.96.1-16.k.1.1, 16.96.1-16.k.1.2, 16.96.1-16.k.1.3, 16.96.1-16.k.1.4, 16.96.1-16.k.1.5, 16.96.1-16.k.1.6, 16.96.1-16.k.1.7, 16.96.1-16.k.1.8, 48.96.1-16.k.1.1, 48.96.1-16.k.1.2, 48.96.1-16.k.1.3, 48.96.1-16.k.1.4, 48.96.1-16.k.1.5, 48.96.1-16.k.1.6, 48.96.1-16.k.1.7, 48.96.1-16.k.1.8, 80.96.1-16.k.1.1, 80.96.1-16.k.1.2, 80.96.1-16.k.1.3, 80.96.1-16.k.1.4, 80.96.1-16.k.1.5, 80.96.1-16.k.1.6, 80.96.1-16.k.1.7, 80.96.1-16.k.1.8, 112.96.1-16.k.1.1, 112.96.1-16.k.1.2, 112.96.1-16.k.1.3, 112.96.1-16.k.1.4, 112.96.1-16.k.1.5, 112.96.1-16.k.1.6, 112.96.1-16.k.1.7, 112.96.1-16.k.1.8, 176.96.1-16.k.1.1, 176.96.1-16.k.1.2, 176.96.1-16.k.1.3, 176.96.1-16.k.1.4, 176.96.1-16.k.1.5, 176.96.1-16.k.1.6, 176.96.1-16.k.1.7, 176.96.1-16.k.1.8, 208.96.1-16.k.1.1, 208.96.1-16.k.1.2, 208.96.1-16.k.1.3, 208.96.1-16.k.1.4, 208.96.1-16.k.1.5, 208.96.1-16.k.1.6, 208.96.1-16.k.1.7, 208.96.1-16.k.1.8, 240.96.1-16.k.1.1, 240.96.1-16.k.1.2, 240.96.1-16.k.1.3, 240.96.1-16.k.1.4, 240.96.1-16.k.1.5, 240.96.1-16.k.1.6, 240.96.1-16.k.1.7, 240.96.1-16.k.1.8, 272.96.1-16.k.1.1, 272.96.1-16.k.1.2, 272.96.1-16.k.1.3, 272.96.1-16.k.1.4, 272.96.1-16.k.1.5, 272.96.1-16.k.1.6, 272.96.1-16.k.1.7, 272.96.1-16.k.1.8, 304.96.1-16.k.1.1, 304.96.1-16.k.1.2, 304.96.1-16.k.1.3, 304.96.1-16.k.1.4, 304.96.1-16.k.1.5, 304.96.1-16.k.1.6, 304.96.1-16.k.1.7, 304.96.1-16.k.1.8
Cyclic 16-isogeny field degree: $8$
Cyclic 16-torsion field degree: $64$
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} - 11x + 14 $
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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
256.a1 $-8$$8000$ $= 2^{6} \cdot 5^{3}$$8.987$$(1:-2:1)$, $(1:2:1)$, $(2:0:1)$, $(0:1:0)$

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 2^6\,\frac{125x^{2}y^{20}+127744x^{2}y^{19}z-5656764x^{2}y^{18}z^{2}-12597072x^{2}y^{17}z^{3}+11614846095x^{2}y^{16}z^{4}-110343749632x^{2}y^{15}z^{5}-8424950796336x^{2}y^{14}z^{6}+3351082327328x^{2}y^{13}z^{7}+2776649776922100x^{2}y^{12}z^{8}+22022464120888064x^{2}y^{11}z^{9}-290271585200163600x^{2}y^{10}z^{10}-4856223435740641104x^{2}y^{9}z^{11}-4834054836258291929x^{2}y^{8}z^{12}+322500259422975321600x^{2}y^{7}z^{13}+1997183810921769840600x^{2}y^{6}z^{14}-4332359677756002799008x^{2}y^{5}z^{15}-84342253426234628895631x^{2}y^{4}z^{16}-213734331495075637624832x^{2}y^{3}z^{17}+794190137019993497305860x^{2}y^{2}z^{18}+5022582383460092582294800x^{2}yz^{19}+7392739626485127577600125x^{2}z^{20}+27040xy^{20}z-1475504xy^{19}z^{2}+92441787xy^{18}z^{3}-619760224xy^{17}z^{4}-116233484298xy^{16}z^{5}+852009573840xy^{15}z^{6}+68860405995848xy^{14}z^{7}+162899771804896xy^{13}z^{8}-17109661499380560xy^{12}z^{9}-165630987617858240xy^{11}z^{10}+1236440858718259865xy^{10}z^{11}+26392226477455615360xy^{9}z^{12}+57751471606219225322xy^{8}z^{13}-1404801279020053973552xy^{7}z^{14}-9905877400591293580108xy^{6}z^{15}+11713037661409532192352xy^{5}z^{16}+349355041719449914480888xy^{4}z^{17}+970601667264738206676240xy^{3}z^{18}-2816276634940196603754875xy^{2}z^{19}-19228590633110988652549600xyz^{20}-28302564912221680304127750xz^{21}+2800y^{21}z+138342y^{20}z^{2}+9609792y^{19}z^{3}-1090691304y^{18}z^{4}+10840708144y^{17}z^{5}+974931906566y^{16}z^{6}-3647510562816y^{15}z^{7}-428389457217952y^{14}z^{8}-2447195613674112y^{13}z^{9}+67055111409348632y^{12}z^{10}+876819015169788160y^{11}z^{11}-1366813727156493760y^{10}z^{12}-83305119373414175952y^{9}z^{13}-376901319134598795382y^{8}z^{14}+2383031506664116281472y^{7}z^{15}+25480697146255442256464y^{6}z^{16}+26090043019966759394032y^{5}z^{17}-504203047045886236972610y^{4}z^{18}-1948004909278555539778496y^{3}z^{19}+1187399128081589212048600y^{2}z^{20}+18366851732381606975917200yz^{21}+27034171318502850297855125z^{22}}{x^{2}y^{20}-122092x^{2}y^{18}z^{2}+463446779x^{2}y^{16}z^{4}-332074040176x^{2}y^{14}z^{6}+86780227553796x^{2}y^{12}z^{8}-11067705526994512x^{2}y^{10}z^{10}+786458406533030771x^{2}y^{8}z^{12}-32838630963742264008x^{2}y^{6}z^{14}+802516752793826493445x^{2}y^{4}z^{16}-10638399653278772101164x^{2}y^{2}z^{18}+59141917011881020620801x^{2}z^{20}-96xy^{20}z+2466455xy^{18}z^{3}-4927450706xy^{16}z^{5}+2506880603368xy^{14}z^{7}-533375068283792xy^{12}z^{9}+59229750777130317xy^{10}z^{11}-3804829914557164622xy^{8}z^{13}+147002042441122410436xy^{6}z^{15}-3376088352702261622824xy^{4}z^{17}+42522117219167233900457xy^{2}z^{19}-226420519297773442433022xz^{21}+4318y^{20}z^{2}-37169800y^{18}z^{4}+41118424318y^{16}z^{6}-14085630594080y^{14}z^{8}+2205765290447864y^{12}z^{10}-188370204625195328y^{10}z^{12}+9513759964055241170y^{8}z^{14}-291674019631871156080y^{6}z^{16}+5314566574799891456790y^{4}z^{18}-52637784574970019446472y^{2}z^{20}+216273370548022802382841z^{22}}$

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
8.24.1.l.1 $8$ $2$ $2$ $1$ $0$ 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
16.96.5.bm.1 $16$ $2$ $2$ $5$ $0$ $1^{4}$
16.96.5.bq.1 $16$ $2$ $2$ $5$ $2$ $1^{4}$
16.96.5.cb.1 $16$ $2$ $2$ $5$ $1$ $1^{4}$
16.96.5.cf.1 $16$ $2$ $2$ $5$ $3$ $1^{4}$
48.96.5.er.1 $48$ $2$ $2$ $5$ $2$ $1^{4}$
48.96.5.et.1 $48$ $2$ $2$ $5$ $0$ $1^{4}$
48.96.5.fn.1 $48$ $2$ $2$ $5$ $3$ $1^{4}$
48.96.5.fp.1 $48$ $2$ $2$ $5$ $1$ $1^{4}$
48.144.9.bo.1 $48$ $3$ $3$ $9$ $7$ $1^{4}\cdot2^{2}$
48.192.9.pa.1 $48$ $4$ $4$ $9$ $3$ $1^{8}$
80.96.5.ev.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ex.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.fl.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.fn.1 $80$ $2$ $2$ $5$ $?$ not computed
80.240.17.u.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.co.1 $80$ $6$ $6$ $17$ $?$ not computed
112.96.5.er.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.et.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.fh.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.fj.1 $112$ $2$ $2$ $5$ $?$ not computed
176.96.5.er.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.et.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.fh.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.fj.1 $176$ $2$ $2$ $5$ $?$ not computed
208.96.5.ev.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.ex.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.fl.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.fn.1 $208$ $2$ $2$ $5$ $?$ not computed
240.96.5.nr.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.nt.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ox.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.oz.1 $240$ $2$ $2$ $5$ $?$ not computed
272.96.5.ev.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.ex.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.fl.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.fn.1 $272$ $2$ $2$ $5$ $?$ not computed
304.96.5.er.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.et.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.fh.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.fj.1 $304$ $2$ $2$ $5$ $?$ not computed