↑

Properties

Label 40.48.1.dl.1
Level $40$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

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 $2^{2}\cdot4$
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: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.340

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&4\\39&3\end{bmatrix}$, $\begin{bmatrix}27&24\\12&31\end{bmatrix}$, $\begin{bmatrix}31&4\\26&27\end{bmatrix}$, $\begin{bmatrix}31&16\\24&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.dl.1.1, 40.96.1-40.dl.1.2, 40.96.1-40.dl.1.3, 40.96.1-40.dl.1.4, 80.96.1-40.dl.1.1, 80.96.1-40.dl.1.2, 80.96.1-40.dl.1.3, 80.96.1-40.dl.1.4, 120.96.1-40.dl.1.1, 120.96.1-40.dl.1.2, 120.96.1-40.dl.1.3, 120.96.1-40.dl.1.4, 240.96.1-40.dl.1.1, 240.96.1-40.dl.1.2, 240.96.1-40.dl.1.3, 240.96.1-40.dl.1.4, 280.96.1-40.dl.1.1, 280.96.1-40.dl.1.2, 280.96.1-40.dl.1.3, 280.96.1-40.dl.1.4
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
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 $ $=$ $ x^{2} + x z + 5 y^{2} - z^{2} $
$=$ $8 x^{2} + 3 x z - 5 y^{2} - 3 z^{2} + w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 81 x^{4} - 22 x^{2} y^{2} + 135 x^{2} z^{2} + y^{4} - 10 y^{2} z^{2} + 25 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 9y$
$\displaystyle Z$ $=$ $\displaystyle \frac{3}{5}w$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4\cdot3^3}{5^2}\cdot\frac{2136706000000xz^{11}-2948328000000xz^{9}w^{2}+1478324520000xz^{7}w^{4}-319202856000xz^{5}w^{6}+27085120200xz^{3}w^{8}-694416240xzw^{10}-1027861000000z^{12}+1589913400000z^{10}w^{2}-939471210000z^{8}w^{4}+260990100000z^{6}w^{6}-33433908300z^{4}w^{8}+1672661340z^{2}w^{10}-20253807w^{12}}{21367060000xz^{11}+455535000xz^{9}w^{2}-819468900xz^{7}w^{4}-78177960xz^{5}w^{6}+3359232xz^{3}w^{8}+708588xzw^{10}-10278610000z^{12}+1453027000z^{10}w^{2}+519256575z^{8}w^{4}-11983950z^{6}w^{6}-8388603z^{4}w^{8}-669222z^{2}w^{10}-59049w^{12}}$

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
8.24.1.m.1 $8$ $2$ $2$ $1$ $0$ dimension zero
20.24.0.d.1 $20$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.w.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.eo.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.ep.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.bm.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.bn.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.gr.1 $40$ $5$ $5$ $17$ $3$ $1^{14}\cdot2$
40.288.17.qj.1 $40$ $6$ $6$ $17$ $2$ $1^{14}\cdot2$
40.480.33.bbl.1 $40$ $10$ $10$ $33$ $6$ $1^{28}\cdot2^{2}$
80.96.3.ic.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.ig.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.iy.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.jc.1 $80$ $2$ $2$ $3$ $?$ not computed
120.144.9.cun.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.bax.1 $120$ $4$ $4$ $9$ $?$ not computed
240.96.3.xa.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.xe.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.zc.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.zg.1 $240$ $2$ $2$ $3$ $?$ not computed