↑

Properties

Label 40.96.1.cd.2
Level $40$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $800$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $2^{6}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.96.1.387

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}9&24\\8&15\end{bmatrix}$, $\begin{bmatrix}11&16\\10&9\end{bmatrix}$, $\begin{bmatrix}21&16\\22&31\end{bmatrix}$, $\begin{bmatrix}33&16\\26&33\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.cd.2.1, 40.192.1-40.cd.2.2, 40.192.1-40.cd.2.3, 40.192.1-40.cd.2.4, 40.192.1-40.cd.2.5, 40.192.1-40.cd.2.6, 40.192.1-40.cd.2.7, 40.192.1-40.cd.2.8, 80.192.1-40.cd.2.1, 80.192.1-40.cd.2.2, 80.192.1-40.cd.2.3, 80.192.1-40.cd.2.4, 80.192.1-40.cd.2.5, 80.192.1-40.cd.2.6, 80.192.1-40.cd.2.7, 80.192.1-40.cd.2.8, 80.192.1-40.cd.2.9, 80.192.1-40.cd.2.10, 80.192.1-40.cd.2.11, 80.192.1-40.cd.2.12, 120.192.1-40.cd.2.1, 120.192.1-40.cd.2.2, 120.192.1-40.cd.2.3, 120.192.1-40.cd.2.4, 120.192.1-40.cd.2.5, 120.192.1-40.cd.2.6, 120.192.1-40.cd.2.7, 120.192.1-40.cd.2.8, 240.192.1-40.cd.2.1, 240.192.1-40.cd.2.2, 240.192.1-40.cd.2.3, 240.192.1-40.cd.2.4, 240.192.1-40.cd.2.5, 240.192.1-40.cd.2.6, 240.192.1-40.cd.2.7, 240.192.1-40.cd.2.8, 240.192.1-40.cd.2.9, 240.192.1-40.cd.2.10, 240.192.1-40.cd.2.11, 240.192.1-40.cd.2.12, 280.192.1-40.cd.2.1, 280.192.1-40.cd.2.2, 280.192.1-40.cd.2.3, 280.192.1-40.cd.2.4, 280.192.1-40.cd.2.5, 280.192.1-40.cd.2.6, 280.192.1-40.cd.2.7, 280.192.1-40.cd.2.8
Cyclic 40-isogeny field degree: $6$
Cyclic 40-torsion field degree: $48$
Full 40-torsion field degree: $7680$

Jacobian

Conductor: $2^{5}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 800.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x y + x z + z w $
$=$ $5 x^{2} + 2 y^{2} + 2 y z - 4 y w - 2 z^{2} - 2 z w + w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} - 8 x^{3} y + 8 x^{3} z + 2 x^{2} y^{2} - 12 x^{2} y z + x^{2} z^{2} + 2 x y^{2} z - 6 x y z^{2} + \cdots - z^{4} $
 Copy content Toggle raw display

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle z$

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{1}{5^2}\cdot\frac{1691894531250xz^{23}-16738378906250xz^{21}w^{2}+99801171875000xz^{19}w^{4}-375547359375000xz^{17}w^{6}+935161125000000xz^{15}w^{8}-1549268785000000xz^{13}w^{10}+1525317724000000xz^{11}w^{12}-673314053600000xz^{9}w^{14}+117074856000000xz^{7}w^{16}-7249100608000xz^{5}w^{18}+133089843200xz^{3}w^{20}-449198080xzw^{22}-757080078125y^{2}z^{22}+9518144531250y^{2}z^{20}w^{2}-47124335937500y^{2}z^{18}w^{4}+177548559375000y^{2}z^{16}w^{6}-401635312500000y^{2}z^{14}w^{8}+622782125000000y^{2}z^{12}w^{10}-515631559600000y^{2}z^{10}w^{12}+173380128480000y^{2}z^{8}w^{14}-21275040416000y^{2}z^{6}w^{16}+845512768000y^{2}z^{4}w^{18}-8056581120y^{2}z^{2}w^{20}+25016320y^{2}w^{22}-757080078125yz^{23}+2190917968750yz^{22}w+9518144531250yz^{21}w^{2}-24584570312500yz^{20}w^{3}-47124335937500yz^{19}w^{4}+123925609375000yz^{18}w^{5}+177548559375000yz^{17}w^{6}-456708506250000yz^{16}w^{7}-401635312500000yz^{15}w^{8}+1017774695000000yz^{14}w^{9}+622782125000000yz^{13}w^{10}-1550839130000000yz^{12}w^{11}-515631559600000yz^{11}w^{12}+1235215471200000yz^{10}w^{13}+173380128480000yz^{9}w^{14}-396810561600000yz^{8}w^{15}-21275040416000yz^{7}w^{16}+46711498176000yz^{6}w^{17}+845512768000yz^{5}w^{18}-1785916646400yz^{4}w^{19}-8056581120yz^{3}w^{20}+16860968960yz^{2}w^{21}+25016320yzw^{22}-21647360yw^{23}+1007080078125z^{24}+1095458984375z^{23}w-10428740234375z^{22}w^{2}-12292285156250z^{21}w^{3}+69534804687500z^{20}w^{4}+61962804687500z^{19}w^{5}-274383629687500z^{18}w^{6}-228354253125000z^{17}w^{7}+755094787500000z^{16}w^{8}+508887347500000z^{15}w^{9}-1363588727500000z^{14}w^{10}-775419565000000z^{13}w^{11}+1583497671600000z^{12}w^{12}+617607735600000z^{11}w^{13}-912698581680000z^{10}w^{14}-198405280800000z^{9}w^{15}+220931213856000z^{8}w^{16}+23355749088000z^{7}w^{17}-20451159776000z^{6}w^{18}-892958323200z^{5}w^{19}+633674603520z^{4}w^{20}+8430484480z^{3}w^{21}-4978048000z^{2}w^{22}-10823680zw^{23}+4194304w^{24}}{w^{4}z^{4}(2707031250xz^{15}-15684843750xz^{13}w^{2}+23654125000xz^{11}w^{4}-10204275000xz^{9}w^{6}+1119020000xz^{7}w^{8}-19748000xz^{5}w^{10}+201600xz^{3}w^{12}+22400xzw^{14}-1531328125y^{2}z^{14}+7393906250y^{2}z^{12}w^{2}-8369862500y^{2}z^{10}w^{4}+2405365000y^{2}z^{8}w^{6}-142782000y^{2}z^{6}w^{8}+1948000y^{2}z^{4}w^{10}+288320y^{2}z^{2}w^{12}+30592y^{2}w^{14}-1531328125yz^{15}+4145468750yz^{14}w+7393906250yz^{13}w^{2}-18970437500yz^{12}w^{3}-8369862500yz^{11}w^{4}+20091525000yz^{10}w^{5}+2405365000yz^{9}w^{6}-5394410000yz^{8}w^{7}-142782000yz^{7}w^{8}+306788000yz^{6}w^{9}+1948000yz^{5}w^{10}-1761600yz^{4}w^{11}+288320yz^{3}w^{12}-62080yz^{2}w^{13}+30592yzw^{14}-8960yw^{15}+1531328125z^{16}+2072734375z^{15}w-11090859375z^{14}w^{2}-9485218750z^{13}w^{3}+22666237500z^{12}w^{4}+10045762500z^{11}w^{5}-14456352500z^{10}w^{6}-2697205000z^{9}w^{7}+2640062000z^{8}w^{8}+153394000z^{7}w^{9}-107202000z^{6}w^{10}-880800z^{5}w^{11}+425280z^{4}w^{12}-31040z^{3}w^{13}+1088z^{2}w^{14}-4480zw^{15})}$

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
40.48.0.k.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.l.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.bc.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.be.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bf.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bu.1 $40$ $2$ $2$ $1$ $1$ 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
40.480.33.lh.1 $40$ $5$ $5$ $33$ $4$ $1^{14}\cdot2^{9}$
40.576.33.wl.1 $40$ $6$ $6$ $33$ $4$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.bhb.2 $40$ $10$ $10$ $65$ $6$ $1^{28}\cdot2^{10}\cdot4^{4}$
80.192.5.dc.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.dn.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.fz.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ga.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.gh.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.gi.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.gt.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.gu.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.hb.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.hc.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.hl.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.hq.2 $80$ $2$ $2$ $5$ $?$ not computed
120.288.17.jvp.1 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.dsf.1 $120$ $4$ $4$ $17$ $?$ not computed
240.192.5.sy.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ts.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.zl.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.zm.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.zt.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.zu.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.baf.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bag.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ban.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bao.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdz.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bep.2 $240$ $2$ $2$ $5$ $?$ not computed