Invariants
Level: | $40$ | $\SL_2$-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: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.399 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&2\\33&11\end{bmatrix}$, $\begin{bmatrix}9&26\\38&3\end{bmatrix}$, $\begin{bmatrix}19&34\\32&9\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $384$ |
Full 40-torsion field degree: | $15360$ |
Jacobian
Conductor: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x^{2} + 3 y^{2} - 2 y z + 2 z^{2} $ |
$=$ | $5 x^{2} - 4 y^{2} + 6 y z - 6 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 12 x^{2} y^{2} - 20 x^{2} z^{2} + 196 y^{4} + 420 y^{2} z^{2} + 225 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{233766000000yz^{11}+1167075000000yz^{9}w^{2}-4842127080000yz^{7}w^{4}+3116065820000yz^{5}w^{6}+591905564600yz^{3}w^{8}-515529514500yzw^{10}-535221000000z^{12}+1406295000000z^{10}w^{2}+1881774090000z^{8}w^{4}-4862237660000z^{6}w^{6}+2202519174100z^{4}w^{8}-87861473700z^{2}w^{10}-49147147049w^{12}}{2164500000yz^{11}-917000000yz^{9}w^{2}+959420000yz^{7}w^{4}-482601000yz^{5}w^{6}+152103350yz^{3}w^{8}+5546310yzw^{10}-4955750000z^{12}+5605100000z^{10}w^{2}-3307535000z^{8}w^{4}+1037575000z^{6}w^{6}-138717775z^{4}w^{8}-25834760z^{2}w^{10}-151263w^{12}}$ |
Modular covers
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 |
---|---|---|---|---|---|---|
8.24.1.bf.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.0.cz.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dd.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dz.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.ei.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.bf.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.1.bs.1 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
40.240.17.ll.1 | $40$ | $5$ | $5$ | $17$ | $6$ | $1^{14}\cdot2$ |
40.288.17.bdl.1 | $40$ | $6$ | $6$ | $17$ | $7$ | $1^{14}\cdot2$ |
40.480.33.btt.1 | $40$ | $10$ | $10$ | $33$ | $11$ | $1^{28}\cdot2^{2}$ |
80.96.3.pa.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.pc.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.te.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.tg.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.144.9.fjb.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.bul.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
240.96.3.bte.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.btg.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.bvu.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.bvw.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |