| $\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}3&28\\36&25\end{bmatrix}$, $\begin{bmatrix}25&32\\18&15\end{bmatrix}$, $\begin{bmatrix}27&0\\18&17\end{bmatrix}$, $\begin{bmatrix}37&4\\38&17\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
40.192.1-40.y.2.1, 40.192.1-40.y.2.2, 40.192.1-40.y.2.3, 40.192.1-40.y.2.4, 40.192.1-40.y.2.5, 40.192.1-40.y.2.6, 40.192.1-40.y.2.7, 40.192.1-40.y.2.8, 80.192.1-40.y.2.1, 80.192.1-40.y.2.2, 80.192.1-40.y.2.3, 80.192.1-40.y.2.4, 80.192.1-40.y.2.5, 80.192.1-40.y.2.6, 80.192.1-40.y.2.7, 80.192.1-40.y.2.8, 80.192.1-40.y.2.9, 80.192.1-40.y.2.10, 80.192.1-40.y.2.11, 80.192.1-40.y.2.12, 120.192.1-40.y.2.1, 120.192.1-40.y.2.2, 120.192.1-40.y.2.3, 120.192.1-40.y.2.4, 120.192.1-40.y.2.5, 120.192.1-40.y.2.6, 120.192.1-40.y.2.7, 120.192.1-40.y.2.8, 240.192.1-40.y.2.1, 240.192.1-40.y.2.2, 240.192.1-40.y.2.3, 240.192.1-40.y.2.4, 240.192.1-40.y.2.5, 240.192.1-40.y.2.6, 240.192.1-40.y.2.7, 240.192.1-40.y.2.8, 240.192.1-40.y.2.9, 240.192.1-40.y.2.10, 240.192.1-40.y.2.11, 240.192.1-40.y.2.12, 280.192.1-40.y.2.1, 280.192.1-40.y.2.2, 280.192.1-40.y.2.3, 280.192.1-40.y.2.4, 280.192.1-40.y.2.5, 280.192.1-40.y.2.6, 280.192.1-40.y.2.7, 280.192.1-40.y.2.8 |
| Cyclic 40-isogeny field degree: |
$12$ |
| Cyclic 40-torsion field degree: |
$96$ |
| Full 40-torsion field degree: |
$7680$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + x y - 2 x w - 2 y^{2} - 2 y w + 2 w^{2} $ |
| $=$ | $4 x^{2} - 2 x y + 4 x w - y^{2} + 4 y w + z^{2} - 4 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 225 x^{4} + 40 x^{2} y^{2} + 40 x^{2} y z - 20 x^{2} z^{2} - y^{4} + 8 y^{3} z - 4 y z^{3} + z^{4} $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 5y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 10w$ |
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{2}{5}\cdot\frac{1212479712xz^{22}w+3093420424320xz^{20}w^{3}+2400192336705600xz^{18}w^{5}+873880102583232000xz^{16}w^{7}+178178945640826240000xz^{14}w^{9}+22226075907781235200000xz^{12}w^{11}+1772983581161896992000000xz^{10}w^{13}+91809011219635523840000000xz^{8}w^{15}+3050380562116795235200000000xz^{6}w^{17}+62112250493638506816000000000xz^{4}w^{19}+694849059523658935200000000000xz^{2}w^{21}+3186149859973388102400000000000xw^{23}+11128464y^{2}z^{22}+89660911440y^{2}z^{20}w^{2}+123442803942000y^{2}z^{18}w^{4}+67705939567072000y^{2}z^{16}w^{6}+19307528982420480000y^{2}z^{14}w^{8}+3240318398300075200000y^{2}z^{12}w^{10}+340888778307610104000000y^{2}z^{10}w^{12}+23105876374621444800000000y^{2}z^{8}w^{14}+1007774187615509944000000000y^{2}z^{6}w^{16}+27310076841456551592000000000y^{2}z^{4}w^{18}+418182169121507188440000000000y^{2}z^{2}w^{20}+2763398413879725312000000000000y^{2}w^{22}+1212479712yz^{22}w+3095809772640yz^{20}w^{3}+2406106612972800yz^{18}w^{5}+878321049024984000yz^{16}w^{7}+179736797296101760000yz^{14}w^{9}+22530043778829539200000yz^{12}w^{11}+1808901386764645824000000yz^{10}w^{13}+94481099826791717600000000yz^{8}w^{15}+3176212274480605923200000000yz^{6}w^{17}+65749252334361287664000000000yz^{4}w^{19}+753692824565147308800000000000yz^{2}w^{21}+3593973951719990114400000000000yw^{23}+2225693z^{24}+17914376736z^{22}w^{2}+24544151152080z^{20}w^{4}+13341337073042400z^{18}w^{6}+3751426280702822000z^{16}w^{8}+616560337910054400000z^{14}w^{10}+62874405086902833600000z^{12}w^{12}+4061755396016921328000000z^{10}w^{14}+163547015574553912700000000z^{8}w^{16}+3800799721911336764800000000z^{6}w^{18}+38533677333449874984000000000z^{4}w^{20}-139123256389859163600000000000z^{2}w^{22}-4578548405077126108600000000000w^{24}}{z^{4}(198720xz^{18}w+391408224xz^{16}w^{3}+221941307600xz^{14}w^{5}+56187244715600xz^{12}w^{7}+7561641444652000xz^{10}w^{9}+585155110608420000xz^{8}w^{11}+26661600993223350000xz^{6}w^{13}+697375511037138000000xz^{4}w^{15}+9489527071938715000000xz^{2}w^{17}+50140248070461800000000xw^{19}+2025y^{2}z^{18}+13124700y^{2}z^{16}w^{2}+13718154850y^{2}z^{14}w^{4}+5433793106000y^{2}z^{12}w^{6}+1064774037905250y^{2}z^{10}w^{8}+116022663132947500y^{2}z^{8}w^{10}+7364494961102662500y^{2}z^{6}w^{12}+270536302413112500000y^{2}z^{4}w^{14}+5327401357486566250000y^{2}z^{2}w^{16}+43487434075246975000000y^{2}w^{18}+198720yz^{18}w+391799184yz^{16}w^{3}+222684473360yz^{14}w^{5}+56591701483600yz^{12}w^{7}+7659065419378000yz^{10}w^{9}+597459010230370000yz^{8}w^{11}+27534962157182900000yz^{6}w^{13}+732412631281204250000yz^{4}w^{15}+10230554610517175000000yz^{2}w^{17}+56558151191144325000000yw^{19}+405z^{20}+2621700z^{18}w^{2}+2722475714z^{16}w^{4}+1064515634400z^{14}w^{6}+204101697037500z^{12}w^{8}+21462770629240500z^{10}w^{10}+1282467757671750000z^{8}w^{12}+41984903173659125000z^{6}w^{14}+619056967076055281250z^{4}w^{16}-112497212923277500000z^{2}w^{18}-72052339947092687500000w^{20})}$ |
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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.
This modular curve is minimally covered by the modular curves in the database listed below.
