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Properties

Label 21.112.1-7.a.1.1
Level $21$
Index $112$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $21$ $\SL_2$-level: $7$ Newform level: $49$
Index: $112$ $\PSL_2$-index:$56$
Genus: $1 = 1 + \frac{ 56 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $7^{8}$ Cusp orbits $1^{2}\cdot6$
Elliptic points: $0$ of order $2$ and $2$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 7B1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 21.112.1.1

Level structure

$\GL_2(\Z/21\Z)$-generators: $\begin{bmatrix}1&7\\14&12\end{bmatrix}$, $\begin{bmatrix}5&7\\7&9\end{bmatrix}$
$\GL_2(\Z/21\Z)$-subgroup: $\SL(2,3):C_6^2$
Contains $-I$: no $\quad$ (see 7.56.1.a.1 for the level structure with $-I$)
Cyclic 21-isogeny field degree: $4$
Cyclic 21-torsion field degree: $48$
Full 21-torsion field degree: $864$

Jacobian

Conductor: $7^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 49.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} + x y $ $=$ $ x^{3} - x^{2} - 2x - 1 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(2:-1:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 56 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{160x^{2}y^{31}-3769x^{2}y^{30}z+151118x^{2}y^{29}z^{2}-1703811x^{2}y^{28}z^{3}+15790688x^{2}y^{27}z^{4}-3330254x^{2}y^{26}z^{5}-1007035907x^{2}y^{25}z^{6}+15073089499x^{2}y^{24}z^{7}-80768574545x^{2}y^{23}z^{8}+282326008368x^{2}y^{22}z^{9}+2338082640537x^{2}y^{21}z^{10}-12343542694139x^{2}y^{20}z^{11}+78053325396479x^{2}y^{19}z^{12}+206379653361932x^{2}y^{18}z^{13}-988475248231674x^{2}y^{17}z^{14}+4117025040474350x^{2}y^{16}z^{15}+20919927604395880x^{2}y^{15}z^{16}+20738226906089523x^{2}y^{14}z^{17}+169106264006470662x^{2}y^{13}z^{18}+775967520102537157x^{2}y^{12}z^{19}+2173581903058954518x^{2}y^{11}z^{20}+7013964635389027050x^{2}y^{10}z^{21}+20807741134395760862x^{2}y^{9}z^{22}+53091120183878325979x^{2}y^{8}z^{23}+123900952402110453903x^{2}y^{7}z^{24}+254305480437400210096x^{2}y^{6}z^{25}+446959743001095182364x^{2}y^{5}z^{26}+655942096275103710504x^{2}y^{4}z^{27}+772586754585347148912x^{2}y^{3}z^{28}+685067983552231108992x^{2}y^{2}z^{29}+417645293426998897536x^{2}yz^{30}+128235728795486150016x^{2}z^{31}+20xy^{32}+786xy^{31}z+15760xy^{30}z^{2}-571235xy^{29}z^{3}+12528793xy^{28}z^{4}-116998157xy^{27}z^{5}+890169934xy^{26}z^{6}+797766137xy^{25}z^{7}-32644636412xy^{24}z^{8}+467624081417xy^{23}z^{9}-1477659907084xy^{22}z^{10}+696158692526xy^{21}z^{11}+58503241693695xy^{20}z^{12}-224656203872388xy^{19}z^{13}+340199914784424xy^{18}z^{14}+5975549361751393xy^{17}z^{15}-1064145101843315xy^{16}z^{16}+10906406040845741xy^{15}z^{17}+202116348025052094xy^{14}z^{18}+471120387059410757xy^{13}z^{19}+1466473793414717699xy^{12}z^{20}+5847640819817387844xy^{11}z^{21}+16660299508759834979xy^{10}z^{22}+44491160061268984343xy^{9}z^{23}+111668991671748548684xy^{8}z^{24}+243475249785175983273xy^{7}z^{25}+465497333303811816978xy^{6}z^{26}+762933165460817924484xy^{5}z^{27}+1034616020539022710704xy^{4}z^{28}+1124146213276275969696xy^{3}z^{29}+919695553612850711520xy^{2}z^{30}+515896477671575109888xyz^{31}+147205565841014140032xz^{32}+y^{33}+570y^{32}z-11220y^{31}z^{2}+143306y^{30}z^{3}+1372869y^{29}z^{4}-30363355y^{28}z^{5}+524058609y^{27}z^{6}-3307215940y^{26}z^{7}+13691635820y^{25}z^{8}+65035350044y^{24}z^{9}-951860920607y^{23}z^{10}+5455123503550y^{22}z^{11}-3301663490044y^{21}z^{12}-67721574848437y^{20}z^{13}+707731026198019y^{19}z^{14}+602710212504006y^{18}z^{15}-3060975529734271y^{17}z^{16}+22100304922212859y^{16}z^{17}+68596274451112210y^{15}z^{18}+137283382128528406y^{14}z^{19}+739181070437670382y^{13}z^{20}+2400873259312608569y^{12}z^{21}+6755646539903243968y^{11}z^{22}+19317345087063425023y^{10}z^{23}+47684447575327705859y^{9}z^{24}+105132824953586659480y^{8}z^{25}+204958394201657166624y^{7}z^{26}+340058616903183005406y^{6}z^{27}+475591444159613338908y^{5}z^{28}+543140720602103603016y^{4}z^{29}+498740389950501728352y^{3}z^{30}+350887192885663264512y^{2}z^{31}+186511846226476577472yz^{32}+59705261123739687936z^{33}}{z^{2}(15x^{2}y^{29}-604x^{2}y^{28}z+24796x^{2}y^{27}z^{2}-203303x^{2}y^{26}z^{3}-2023761x^{2}y^{25}z^{4}+33078247x^{2}y^{24}z^{5}+10597956x^{2}y^{23}z^{6}-1076337405x^{2}y^{22}z^{7}+1649022883x^{2}y^{21}z^{8}+19127723450x^{2}y^{20}z^{9}-12779467501x^{2}y^{19}z^{10}-81730835960x^{2}y^{18}z^{11}+457690794890x^{2}y^{17}z^{12}+2014081451373x^{2}y^{16}z^{13}+5267422223457x^{2}y^{15}z^{14}+21031565708377x^{2}y^{14}z^{15}+83798601602244x^{2}y^{13}z^{16}+278636593824683x^{2}y^{12}z^{17}+826154427742550x^{2}y^{11}z^{18}+2212263240499720x^{2}y^{10}z^{19}+5201018775347193x^{2}y^{9}z^{20}+10390250973752311x^{2}y^{8}z^{21}+17125528797972961x^{2}y^{7}z^{22}+22653926592588718x^{2}y^{6}z^{23}+23383262878734388x^{2}y^{5}z^{24}+18239968790833888x^{2}y^{4}z^{25}+10313286030007872x^{2}y^{3}z^{26}+3968278124830368x^{2}y^{2}z^{27}+926331000146688x^{2}yz^{28}+98772381246848x^{2}z^{29}+xy^{30}+140xy^{29}z+2139xy^{28}z^{2}-151548xy^{27}z^{3}+1891160xy^{26}z^{4}+5392943xy^{25}z^{5}-143890289xy^{24}z^{6}+200791731xy^{23}z^{7}+4052983121xy^{22}z^{8}-6262216413xy^{21}z^{9}-42835709907xy^{20}z^{10}+130009714944xy^{19}z^{11}+540985931737xy^{18}z^{12}+345154740404xy^{17}z^{13}+2411738235380xy^{16}z^{14}+17591998353899xy^{15}z^{15}+64457747381615xy^{14}z^{16}+206751924294829xy^{13}z^{17}+647813887670400xy^{12}z^{18}+1849847785132091xy^{11}z^{19}+4647792084095424xy^{10}z^{20}+10094725775529282xy^{9}z^{21}+18532566243666947xy^{8}z^{22}+28085930425752301xy^{7}z^{23}+34340972852628932xy^{6}z^{24}+33050406837580720xy^{5}z^{25}+24286019379979016xy^{4}z^{26}+13068638284813408xy^{3}z^{27}+4829901807918496xy^{2}z^{28}+1091431857227712xyz^{29}+113377841165952xz^{30}+77y^{30}z-2163y^{29}z^{2}+7778y^{28}z^{3}+731089y^{27}z^{4}-6987564y^{26}z^{5}-19106424y^{25}z^{6}+391822953y^{24}z^{7}+30445023y^{23}z^{8}-7158961904y^{22}z^{9}+7582107411y^{21}z^{10}+87591886623y^{20}z^{11}+28400121121y^{19}z^{12}-19287748192y^{18}z^{13}+1706293220507y^{17}z^{14}+7533311149214y^{16}z^{15}+24968145810420y^{15}z^{16}+85394829220275y^{14}z^{17}+273220977864475y^{13}z^{18}+783541498655997y^{12}z^{19}+1990618366913414y^{11}z^{20}+4379858537981833y^{10}z^{21}+8151060468848674y^{9}z^{22}+12609847242133754y^{8}z^{23}+16055708218501533y^{7}z^{24}+16739565940085842y^{6}z^{25}+14188461238375576y^{5}z^{26}+9592765436860208y^{4}z^{27}+4968287258673520y^{3}z^{28}+1834689530733440y^{2}z^{29}+425153879888064yz^{30}+45982909828224z^{31})}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
21.16.0-7.a.1.1 $21$ $7$ $7$ $0$ $0$ full Jacobian
21.16.0-7.a.1.2 $21$ $7$ $7$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
21.336.3-7.a.1.1 $21$ $3$ $3$ $3$ $0$ $1^{2}$
21.336.3-7.b.1.1 $21$ $3$ $3$ $3$ $0$ $2$
21.336.3-7.b.1.2 $21$ $3$ $3$ $3$ $0$ $2$
21.336.11-21.a.1.2 $21$ $3$ $3$ $11$ $1$ $1^{2}\cdot2^{4}$
21.448.11-21.a.1.3 $21$ $4$ $4$ $11$ $2$ $1^{6}\cdot2^{2}$
42.224.5-14.a.1.1 $42$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
42.224.5-14.c.1.1 $42$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
42.224.5-42.c.1.1 $42$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
42.224.5-42.h.1.1 $42$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
42.336.7-14.a.1.1 $42$ $3$ $3$ $7$ $0$ $1^{4}\cdot2$
63.336.3-63.a.1.1 $63$ $3$ $3$ $3$ $0$ $2$
63.336.3-63.a.1.2 $63$ $3$ $3$ $3$ $0$ $2$
63.336.3-63.b.1.1 $63$ $3$ $3$ $3$ $0$ $2$
63.336.3-63.b.1.2 $63$ $3$ $3$ $3$ $0$ $2$
63.336.3-63.c.1.1 $63$ $3$ $3$ $3$ $0$ $2$
63.336.3-63.c.1.2 $63$ $3$ $3$ $3$ $0$ $2$
63.3024.113-63.a.1.1 $63$ $27$ $27$ $113$ $52$ $1^{21}\cdot2^{2}\cdot3^{13}\cdot4^{5}\cdot5^{2}\cdot6\cdot12$
84.224.5-28.b.1.1 $84$ $2$ $2$ $5$ $?$ not computed
84.224.5-28.h.1.2 $84$ $2$ $2$ $5$ $?$ not computed
84.224.5-84.h.1.1 $84$ $2$ $2$ $5$ $?$ not computed
84.224.5-84.w.1.1 $84$ $2$ $2$ $5$ $?$ not computed
84.448.15-28.e.1.4 $84$ $4$ $4$ $15$ $?$ not computed
168.224.5-56.b.1.3 $168$ $2$ $2$ $5$ $?$ not computed
168.224.5-56.h.1.3 $168$ $2$ $2$ $5$ $?$ not computed
168.224.5-56.z.1.1 $168$ $2$ $2$ $5$ $?$ not computed
168.224.5-168.z.1.2 $168$ $2$ $2$ $5$ $?$ not computed
168.224.5-56.bf.1.1 $168$ $2$ $2$ $5$ $?$ not computed
168.224.5-168.bf.1.2 $168$ $2$ $2$ $5$ $?$ not computed
168.224.5-168.dh.1.2 $168$ $2$ $2$ $5$ $?$ not computed
168.224.5-168.dn.1.4 $168$ $2$ $2$ $5$ $?$ not computed
210.224.5-70.c.1.2 $210$ $2$ $2$ $5$ $?$ not computed
210.224.5-70.h.1.2 $210$ $2$ $2$ $5$ $?$ not computed
210.224.5-210.h.1.3 $210$ $2$ $2$ $5$ $?$ not computed
210.224.5-210.w.1.1 $210$ $2$ $2$ $5$ $?$ not computed
273.336.3-91.a.1.1 $273$ $3$ $3$ $3$ $?$ not computed
273.336.3-91.a.1.3 $273$ $3$ $3$ $3$ $?$ not computed
273.336.3-91.b.1.1 $273$ $3$ $3$ $3$ $?$ not computed
273.336.3-91.b.1.3 $273$ $3$ $3$ $3$ $?$ not computed
273.336.3-91.c.1.1 $273$ $3$ $3$ $3$ $?$ not computed
273.336.3-91.c.1.3 $273$ $3$ $3$ $3$ $?$ not computed