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Properties

Label 40.48.1.be.2
Level $40$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $800$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.79

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}7&0\\0&33\end{bmatrix}$, $\begin{bmatrix}9&8\\2&27\end{bmatrix}$, $\begin{bmatrix}9&16\\28&23\end{bmatrix}$, $\begin{bmatrix}39&0\\26&7\end{bmatrix}$, $\begin{bmatrix}39&12\\32&33\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.be.2.1, 40.96.1-40.be.2.2, 40.96.1-40.be.2.3, 40.96.1-40.be.2.4, 40.96.1-40.be.2.5, 40.96.1-40.be.2.6, 40.96.1-40.be.2.7, 40.96.1-40.be.2.8, 40.96.1-40.be.2.9, 40.96.1-40.be.2.10, 40.96.1-40.be.2.11, 40.96.1-40.be.2.12, 40.96.1-40.be.2.13, 40.96.1-40.be.2.14, 40.96.1-40.be.2.15, 40.96.1-40.be.2.16, 120.96.1-40.be.2.1, 120.96.1-40.be.2.2, 120.96.1-40.be.2.3, 120.96.1-40.be.2.4, 120.96.1-40.be.2.5, 120.96.1-40.be.2.6, 120.96.1-40.be.2.7, 120.96.1-40.be.2.8, 120.96.1-40.be.2.9, 120.96.1-40.be.2.10, 120.96.1-40.be.2.11, 120.96.1-40.be.2.12, 120.96.1-40.be.2.13, 120.96.1-40.be.2.14, 120.96.1-40.be.2.15, 120.96.1-40.be.2.16, 280.96.1-40.be.2.1, 280.96.1-40.be.2.2, 280.96.1-40.be.2.3, 280.96.1-40.be.2.4, 280.96.1-40.be.2.5, 280.96.1-40.be.2.6, 280.96.1-40.be.2.7, 280.96.1-40.be.2.8, 280.96.1-40.be.2.9, 280.96.1-40.be.2.10, 280.96.1-40.be.2.11, 280.96.1-40.be.2.12, 280.96.1-40.be.2.13, 280.96.1-40.be.2.14, 280.96.1-40.be.2.15, 280.96.1-40.be.2.16
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 275x - 1750 $
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Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{5^2}\cdot\frac{120x^{2}y^{14}+265491250x^{2}y^{12}z^{2}+79026301875000x^{2}y^{10}z^{4}+5747837250087890625x^{2}y^{8}z^{6}+134200360963593750000000x^{2}y^{6}z^{8}+1276514304000111236572265625x^{2}y^{4}z^{10}+5221768888319999771118164062500x^{2}y^{2}z^{12}+7636775567360000002384185791015625x^{2}z^{14}+25900xy^{14}z+20577225000xy^{12}z^{3}+3967774374609375xy^{10}z^{5}+200163304666113281250xy^{8}z^{7}+3697927782368505859375000xy^{6}z^{9}+29987168255998883056640625000xy^{4}z^{11}+109220770611200002384185791015625xy^{2}z^{13}+146184193638399999976158142089843750xz^{15}+y^{16}+2514000y^{14}z^{2}+1359213562500y^{12}z^{4}+153546889734375000y^{10}z^{6}+4917296730081054687500y^{8}z^{8}+61279076351282226562500000y^{6}z^{10}+340493631487980628967285156250y^{4}z^{12}+832083525632000038146972656250000y^{2}z^{14}+698164379647999999582767486572265625z^{16}}{zy^{4}(7175x^{2}y^{8}z+208750000x^{2}y^{6}z^{3}+896023828125x^{2}y^{4}z^{5}+195312500x^{2}y^{2}z^{7}+6103515625x^{2}z^{9}+xy^{10}+298750xy^{8}z^{2}+5082937500xy^{6}z^{4}+17151781250000xy^{4}z^{6}-1708984375xy^{2}z^{8}-61035156250xz^{10}+120y^{10}z+8153750y^{8}z^{3}+60701250000y^{6}z^{5}+81915449218750y^{4}z^{7}-39062500000y^{2}z^{9}-1068115234375z^{11})}$

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
8.24.0.e.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.h.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.c.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
40.96.1.h.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.x.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.bd.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.bh.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.bv.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.bz.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.cb.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.cd.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.240.17.by.2 $40$ $5$ $5$ $17$ $2$ $1^{6}\cdot2^{5}$
40.288.17.eu.1 $40$ $6$ $6$ $17$ $3$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.ix.2 $40$ $10$ $10$ $33$ $3$ $1^{12}\cdot2^{6}\cdot4^{2}$
120.96.1.fw.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.gc.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.hd.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.hj.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.mf.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ml.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.nl.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.nr.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.144.9.rz.2 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.jj.2 $120$ $4$ $4$ $9$ $?$ not computed
280.96.1.fx.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.gb.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.gn.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.gr.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ij.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.in.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.iz.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.jd.1 $280$ $2$ $2$ $1$ $?$ dimension zero