Invariants
| Level: | $32$ | $\SL_2$-level: | $32$ | Newform level: | $128$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $2 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 14 }{2}$ | ||||||
| Cusps: | $14$ (of which $2$ are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot32^{2}$ | Cusp orbits | $1^{2}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 32B2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 32.192.2.74 |
Level structure
| $\GL_2(\Z/32\Z)$-generators: | $\begin{bmatrix}1&7\\0&27\end{bmatrix}$, $\begin{bmatrix}11&5\\8&13\end{bmatrix}$, $\begin{bmatrix}21&18\\8&9\end{bmatrix}$, $\begin{bmatrix}31&2\\16&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 32.96.2.c.1 for the level structure with $-I$) |
| Cyclic 32-isogeny field degree: | $4$ |
| Cyclic 32-torsion field degree: | $16$ |
| Full 32-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{14}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $2$ |
| Newforms: | 128.2.e.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x y w + y w^{2} + z w^{2} $ |
| $=$ | $2 x y^{2} + y^{2} w + y z w$ | |
| $=$ | $2 x y z + y z w + z^{2} w$ | |
| $=$ | $2 x^{2} y + x y w + x z w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{3} z + 2 x^{2} y^{2} - 2 x^{2} z^{2} + x z^{3} + 2 y^{2} z^{2} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 2x^{5} + 4x^{4} + 4x^{2} - 2x $ |
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.
| Embedded model |
|---|
| $(0:1:0:0)$, $(1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{393216x^{20}+524288x^{19}w-69533696x^{18}w^{2}-24838144x^{17}w^{3}+4189618176x^{16}w^{4}-2507702272x^{15}w^{5}-87372648448x^{14}w^{6}+130250575872x^{13}w^{7}+91811241472x^{12}w^{8}-131957102592x^{11}w^{9}-198864824576x^{10}w^{10}+203192050944x^{9}w^{11}+191590849056x^{8}w^{12}-412397628608x^{7}w^{13}+411680460640x^{6}w^{14}-509640717312x^{5}w^{15}+1202736235452x^{4}w^{16}-3090311436616x^{3}w^{17}+2918296751430x^{2}w^{18}+4174685590490xw^{19}-256y^{20}-512y^{18}w^{2}+48864y^{16}w^{4}-1472y^{14}w^{6}-3259296y^{12}w^{8}+6621776y^{10}w^{10}+71823282y^{8}w^{12}-309050344y^{6}w^{14}+308934921y^{4}w^{16}+873727584y^{2}w^{18}-38454311936yz^{19}+111789354496yz^{17}w^{2}+4889334528yz^{15}w^{4}-246542506752yz^{13}w^{6}+43039478016yz^{11}w^{8}+569248590304yz^{9}w^{10}-1150433164608yz^{7}w^{12}+1440952904464yz^{5}w^{14}+506245134196yz^{3}w^{16}-4675736470006yzw^{18}+15923107072z^{20}-27078702080z^{18}w^{2}-76915576800z^{16}w^{4}+202720425024z^{14}w^{6}-209796644960z^{12}w^{8}+733441427248z^{10}w^{10}-2495151613746z^{8}w^{12}+4121669520008z^{6}w^{14}+308725332651z^{4}w^{16}-11496237005222z^{2}w^{18}+3203952796960w^{20}}{w^{2}(98304x^{18}+32768x^{17}w+892928x^{16}w^{2}-999424x^{15}w^{3}+5074944x^{14}w^{4}-11718656x^{13}w^{5}+31814656x^{12}w^{6}-79851520x^{11}w^{7}+194421376x^{10}w^{8}-463731328x^{9}w^{9}+990670368x^{8}w^{10}-2102404672x^{7}w^{11}+4296000440x^{6}w^{12}-8736992168x^{5}w^{13}+17473105834x^{4}w^{14}-32259995600x^{3}w^{15}+41342996225x^{2}w^{16}+53995305241xw^{17}-64y^{16}w^{2}+152y^{12}w^{6}-448y^{10}w^{8}+1608y^{8}w^{10}-5744y^{6}w^{12}+21167y^{4}w^{14}-80144y^{2}w^{16}-214682624yz^{15}w^{2}+540254720yz^{13}w^{4}-648446880yz^{11}w^{6}+550610352yz^{9}w^{8}-360705432yz^{7}w^{10}-781449500yz^{5}w^{12}+9012301258yz^{3}w^{14}-58282043963yzw^{16}+87979072z^{16}w^{2}-114374144z^{14}w^{4}-98509240z^{12}w^{6}+462833712z^{10}w^{8}-597063168z^{8}w^{10}-1885395180z^{6}w^{12}+22189938595z^{4}w^{14}-145203240583z^{2}w^{16}+40317072214w^{18})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 32.96.2.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}Y^{2}-X^{3}Z-2X^{2}Z^{2}+2Y^{2}Z^{2}+XZ^{3} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 32.96.2.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y^{2}w+z^{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.0-16.j.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 32.96.0-16.j.1.1 | $32$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 32.384.5-32.n.1.5 | $32$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 32.384.5-32.n.2.7 | $32$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 32.384.5-32.o.1.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 32.384.5-32.o.2.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 32.384.5-32.s.1.6 | $32$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 32.384.5-32.s.2.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 32.384.5-32.t.1.7 | $32$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 32.384.5-32.t.2.5 | $32$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 32.384.7-32.bc.1.2 | $32$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 32.384.7-32.bd.1.4 | $32$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 32.384.7-32.bp.1.3 | $32$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 32.384.7-32.bq.1.2 | $32$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 96.384.5-96.bb.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bb.2.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bc.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bc.2.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bk.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bk.2.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bl.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bl.2.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.7-96.bu.1.12 | $96$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 96.384.7-96.bv.1.10 | $96$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 96.384.7-96.cb.1.6 | $96$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 96.384.7-96.cc.1.4 | $96$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 160.384.5-160.bs.1.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.bs.2.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.bt.1.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.bt.2.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.ck.1.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.ck.2.14 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cl.1.11 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cl.2.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.7-160.cj.1.11 | $160$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 160.384.7-160.ck.1.11 | $160$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 160.384.7-160.cx.1.3 | $160$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 160.384.7-160.cy.1.3 | $160$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 224.384.5-224.bb.1.11 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bb.2.1 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bc.1.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bc.2.14 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bk.1.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bk.2.14 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bl.1.11 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bl.2.1 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.7-224.bu.1.11 | $224$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 224.384.7-224.bv.1.9 | $224$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 224.384.7-224.cb.1.5 | $224$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 224.384.7-224.cc.1.5 | $224$ | $2$ | $2$ | $7$ | $?$ | not computed |