↑

Properties

Label 40.40.1.u.1
Level $40$
Index $40$
Genus $1$
Analytic rank $1$
Cusps $4$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $40$ $\SL_2$-level: $10$ Newform level: $1600$
Index: $40$ $\PSL_2$-index:$40$
Genus: $1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $10^{4}$ Cusp orbits $4$
Elliptic points: $0$ of order $2$ and $4$ 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: 10H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.40.1.35

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}5&1\\14&29\end{bmatrix}$, $\begin{bmatrix}24&9\\25&11\end{bmatrix}$, $\begin{bmatrix}33&19\\21&32\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 40-isogeny field degree: $72$
Cyclic 40-torsion field degree: $1152$
Full 40-torsion field degree: $18432$

Jacobian

Conductor: $2^{6}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1600.2.a.c

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - 133x + 363 $
 Copy content Toggle raw display

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 40 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^6\cdot5^4\,\frac{17496x^{2}y^{25}+254611440x^{2}y^{24}z+14137305600x^{2}y^{23}z^{2}+3962800320000x^{2}y^{22}z^{3}-685783779200000x^{2}y^{21}z^{4}-20311650400000000x^{2}y^{20}z^{5}+5962480000000000000x^{2}y^{19}z^{6}-171744820160000000000x^{2}y^{18}z^{7}-23034693675200000000000x^{2}y^{17}z^{8}+1588170602960000000000000x^{2}y^{16}z^{9}+23078737689600000000000000x^{2}y^{15}z^{10}-5658796490880000000000000000x^{2}y^{14}z^{11}+118763866080000000000000000000x^{2}y^{13}z^{12}+9783659395520000000000000000000x^{2}y^{12}z^{13}-508698811187200000000000000000000x^{2}y^{11}z^{14}-4333685475200000000000000000000000x^{2}y^{10}z^{15}+893020543873600000000000000000000000x^{2}y^{9}z^{16}-14092071680880000000000000000000000000x^{2}y^{8}z^{17}-746762642880000000000000000000000000000x^{2}y^{7}z^{18}+27903229727680000000000000000000000000000x^{2}y^{6}z^{19}+127564145872000000000000000000000000000000x^{2}y^{5}z^{20}-20432475622240000000000000000000000000000000x^{2}y^{4}z^{21}+235098106803200000000000000000000000000000000x^{2}y^{3}z^{22}+4677883225920000000000000000000000000000000000x^{2}y^{2}z^{23}-126393767496000000000000000000000000000000000000x^{2}yz^{24}+816601543600000000000000000000000000000000000000x^{2}z^{25}+189xy^{26}+17077392xy^{25}z+12674752760xy^{24}z^{2}-622215961600xy^{23}z^{3}-125453496720000xy^{22}z^{4}+15994567443200000xy^{21}z^{5}+199412901200000000xy^{20}z^{6}-113747099648000000000xy^{19}z^{7}+4298909444760000000000xy^{18}z^{8}+354185766371200000000000xy^{17}z^{9}-29800465959960000000000000xy^{16}z^{10}-153816706713600000000000000xy^{15}z^{11}+90812412747680000000000000000xy^{14}z^{12}-2374200590528000000000000000000xy^{13}z^{13}-134947557234720000000000000000000xy^{12}z^{14}+8158009090355200000000000000000000xy^{11}z^{15}+29428986251000000000000000000000000xy^{10}z^{16}-12775491840681600000000000000000000000xy^{9}z^{17}+234185920484680000000000000000000000000xy^{8}z^{18}+9583024812672000000000000000000000000000xy^{7}z^{19}-392248507665680000000000000000000000000000xy^{6}z^{20}-1007478409504000000000000000000000000000000xy^{5}z^{21}+262463374437840000000000000000000000000000000xy^{4}z^{22}-3212951147827200000000000000000000000000000000xy^{3}z^{23}-54721279407320000000000000000000000000000000000xy^{2}z^{24}+1539518680016000000000000000000000000000000000000xyz^{25}-9946484450600000000000000000000000000000000000000xz^{26}+718551y^{26}z+2390370840y^{25}z^{2}-220140887240y^{24}z^{3}+16703302054400y^{23}z^{4}+1552083259280000y^{22}z^{5}-281284844796800000y^{21}z^{6}+4547297058000000000y^{20}z^{7}+1418376024704000000000y^{19}z^{8}-78892397022840000000000y^{18}z^{9}-2474748861316800000000000y^{17}z^{10}+354513408419240000000000000y^{16}z^{11}-4312056783385600000000000000y^{15}z^{12}-761142281449120000000000000000y^{14}z^{13}+30971353497824000000000000000000y^{13}z^{14}+643221820640480000000000000000000y^{12}z^{15}-68327915711180800000000000000000000y^{11}z^{16}+611994944061800000000000000000000000y^{10}z^{17}+74353873057422400000000000000000000000y^{9}z^{18}-2108688190544120000000000000000000000000y^{8}z^{19}-31747873342656000000000000000000000000000y^{7}z^{20}+2135394105787920000000000000000000000000000y^{6}z^{21}-11317045851056000000000000000000000000000000y^{5}z^{22}-901194856861360000000000000000000000000000000y^{4}z^{23}+15334724051532800000000000000000000000000000000y^{3}z^{24}+71594880986680000000000000000000000000000000000y^{2}z^{25}-3481010382984000000000000000000000000000000000000yz^{26}+22490041349400000000000000000000000000000000000000z^{27}}{396400x^{2}y^{24}z-2102865600000x^{2}y^{22}z^{3}+593232560800000000x^{2}y^{20}z^{5}-15660744968000000000000x^{2}y^{18}z^{7}+111855781458000000000000000x^{2}y^{16}z^{9}-389087587708800000000000000000x^{2}y^{14}z^{11}+798924297304000000000000000000000x^{2}y^{12}z^{13}-1052619137424000000000000000000000000x^{2}y^{10}z^{15}+920284268522000000000000000000000000000x^{2}y^{8}z^{17}-533773430552000000000000000000000000000000x^{2}y^{6}z^{19}+198137695005600000000000000000000000000000000x^{2}y^{4}z^{21}-42736328116800000000000000000000000000000000000x^{2}y^{2}z^{23}+4083007812400000000000000000000000000000000000000x^{2}z^{25}+xy^{26}-101071400xy^{24}z^{2}+184955799600000xy^{22}z^{4}-22260006546800000000xy^{20}z^{6}+354117677723000000000000xy^{18}z^{8}-2061045062235000000000000000xy^{16}z^{10}+6366719320928800000000000000000xy^{14}z^{12}-12070559618724000000000000000000000xy^{12}z^{14}+15003137875879000000000000000000000000xy^{10}z^{16}-12540320029967000000000000000000000000000xy^{8}z^{18}+7016117218446000000000000000000000000000000xy^{6}z^{20}-2528181641935600000000000000000000000000000000xy^{4}z^{22}+531828125033800000000000000000000000000000000000xy^{2}z^{24}-49732421875400000000000000000000000000000000000000xz^{26}-933y^{26}z+17301696600y^{24}z^{3}-11972800228400000y^{22}z^{5}+615941671533200000000y^{20}z^{7}-5788602704607000000000000y^{18}z^{9}+24056361480693000000000000000y^{16}z^{11}-57136894629607200000000000000000y^{14}z^{13}+86473521971036000000000000000000000y^{12}z^{15}-87631454505491000000000000000000000000y^{10}z^{17}+60433333435553000000000000000000000000000y^{8}z^{19}-28058844567790000000000000000000000000000000y^{6}z^{21}+8399450231856400000000000000000000000000000000y^{4}z^{23}-1463201172860200000000000000000000000000000000000y^{2}z^{25}+112450195324600000000000000000000000000000000000000z^{27}}$

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
20.20.0.c.1 $20$ $2$ $2$ $0$ $0$ full Jacobian
40.20.0.c.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.20.1.b.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
40.120.5.r.1 $40$ $3$ $3$ $5$ $2$ $1^{4}$
40.120.5.cq.1 $40$ $3$ $3$ $5$ $2$ $1^{4}$
40.160.9.u.1 $40$ $4$ $4$ $9$ $4$ $1^{8}$
120.120.9.fw.1 $120$ $3$ $3$ $9$ $?$ not computed
120.160.9.bg.1 $120$ $4$ $4$ $9$ $?$ not computed
200.200.9.m.1 $200$ $5$ $5$ $9$ $?$ not computed
280.320.21.u.1 $280$ $8$ $8$ $21$ $?$ not computed