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Properties

Label 13.168.2-13.c.2.2
Level $13$
Index $168$
Genus $2$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $13$ $\SL_2$-level: $13$ Newform level: $169$
Index: $168$ $\PSL_2$-index:$84$
Genus: $2 = 1 + \frac{ 84 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $1^{6}\cdot13^{6}$ Cusp orbits $3^{2}\cdot6$
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: 13A2
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 13.168.2.9
Sutherland (S) label: 13B.1.6

Level structure

$\GL_2(\Z/13\Z)$-generators: $\begin{bmatrix}3&2\\0&3\end{bmatrix}$, $\begin{bmatrix}11&10\\0&3\end{bmatrix}$
$\GL_2(\Z/13\Z)$-subgroup: $C_{13}:C_{12}$
Contains $-I$: no $\quad$ (see 13.84.2.c.2 for the level structure with $-I$)
Cyclic 13-isogeny field degree: $1$
Cyclic 13-torsion field degree: $12$
Full 13-torsion field degree: $156$

Jacobian

Conductor: $13^{4}$
Simple: yes
Squarefree: yes
Decomposition: $2$
Newforms: 169.2.b.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x^{2} z - x y z + 3 x z^{2} - x z w + 3 y^{2} z - y z^{2} - 4 y z w $
$=$ $2 x^{2} y - x y^{2} + 3 x y z - x y w + 3 y^{3} - y^{2} z - 4 y^{2} w$
$=$ $2 x^{2} w - x y w + 3 x z w - x w^{2} + 3 y^{2} w - y z w - 4 y w^{2}$
$=$ $2 x^{3} - x^{2} y + 3 x^{2} z - x^{2} w + 3 x y^{2} - x y z - 4 x y w$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 17 x^{5} - 2 x^{4} y + 67 x^{4} z - x^{3} y^{2} - 19 x^{3} y z + 211 x^{3} z^{2} + x^{2} y^{2} z + \cdots + 79 z^{5} $
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Weierstrass model Weierstrass model

$ y^{2} + \left(x^{2} + x\right) y $ $=$ $ 2x^{6} + 9x^{5} + 28x^{4} + 31x^{3} + 13x^{2} + 3x + 2 $
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Rational points

This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 84 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^{12}\,\frac{26785892129684953519935254089728xyz^{16}+43515562187388285393441389273088xyz^{15}w-10313653772060735219680184821735424xyz^{14}w^{2}+21028627670341406690870047490004992xyz^{13}w^{3}+368515109595154417187424680853651456xyz^{12}w^{4}-1227449301852005169856019099409327104xyz^{11}w^{5}-1376747022575391538381659203772761088xyz^{10}w^{6}+9473588274443613493644159133574176384xyz^{9}w^{7}-10316828624480682342502717579747455680xyz^{8}w^{8}-639023639186085829614703411798471712xyz^{7}w^{9}+5744460129704415768204017419073937312xyz^{6}w^{10}-10242224969603269564703594531722138040xyz^{5}w^{11}+24649139215141920360887134003745642528xyz^{4}w^{12}-24170477536530410901491947061653328846xyz^{3}w^{13}+6623165652531906686858150591264589347xyz^{2}w^{14}+1916221037710338169920947361567993461xyzw^{15}-816648101315440743286487450021204866xyw^{16}-16986154679121620393751427526656xz^{17}-249912695595247975483226727641088xz^{16}w+6490732058658069015342663035633664xz^{15}w^{2}+17121269028087662332663755805259776xz^{14}w^{3}-278210482135370869244786997611603968xz^{13}w^{4}+14685974435785151732425381650404352xz^{12}w^{5}+2924786675372128449642311099808160768xz^{11}w^{6}-2954385299003396446315841437840013952xz^{10}w^{7}-8674888101093792473504329658936238656xz^{9}w^{8}+13769655606884885019167954581794564704xz^{8}w^{9}+475701719055935031795508691594570464xz^{7}w^{10}-4239196817658454456806825434669281272xz^{6}w^{11}+3866781905789462634800398773075503656xz^{5}w^{12}-25398222274080380294531266906153153434xz^{4}w^{13}+31054495349087879956993661252823658853xz^{3}w^{14}-8611242992854221073199326780498939556xz^{2}w^{15}-3251550431435420473465482947620183317xzw^{16}+1284203225243009840773286951057781534xw^{17}-22322085390329529992278412562432y^{2}z^{16}+731236242513284813622173930463232y^{2}z^{15}w+6116532467135462882585857690288128y^{2}z^{14}w^{2}-94137242333574121296575425620318208y^{2}z^{13}w^{3}+12663248938591061198584599904581632y^{2}z^{12}w^{4}+1888864407522018872818855586774725632y^{2}z^{11}w^{5}-4479351451331501167416207695986561536y^{2}z^{10}w^{6}-2499446694516090691198620759171079552y^{2}z^{9}w^{7}+19481805791489987582990392723630974912y^{2}z^{8}w^{8}-24979978859719397601189376768609478432y^{2}z^{7}w^{9}+19798951504072086528319185590595167072y^{2}z^{6}w^{10}-20847729567926255424016661666413078680y^{2}z^{5}w^{11}+8896659121488778714094040929479532136y^{2}z^{4}w^{12}+17246260283655535421537173611907111854y^{2}z^{3}w^{13}-21182607823537362762537623871413223149y^{2}z^{2}w^{14}+7009469306318235190360677962933896185y^{2}zw^{15}-259746043341480993600864137394427290y^{2}w^{16}+61996803278925870420029878571008yz^{17}-290145697438235508946121459343360yz^{16}w-22637231928531235538875934275145728yz^{15}w^{2}+85613303741790988491413137751246848yz^{14}w^{3}+717876237669281571225535640182931456yz^{13}w^{4}-3245819672039500336333117239360012288yz^{12}w^{5}-1445315316933574557246513432647978496yz^{11}w^{6}+21765408335652051975203985298090283520yz^{10}w^{7}-25209672742054157985531748927018674624yz^{9}w^{8}-9818024304108477671827628861533210624yz^{8}w^{9}+25970340165356547145053305200632545552yz^{7}w^{10}-20585644978022204965558551422016152840yz^{6}w^{11}+49358088893536612368975591341472628480yz^{5}w^{12}-49843805013163094962038305501587599164yz^{4}w^{13}-9611781918503176068463784582038000365yz^{3}w^{14}+33631018612994933246536532749613011389yz^{2}w^{15}-11747642690214743750317571858776226128yzw^{16}+4311687723365380409958269352129476yw^{17}-30530847241895482503310036500480z^{18}-220535435695742123920410121404416z^{17}w+11123522845524059786679524707659776z^{16}w^{2}+5976776805231149580142078249783296z^{15}w^{3}-429027917696912631606806918358104064z^{14}w^{4}+466850631428318242435119435057439744z^{13}w^{5}+3787985237328251439303909366376516608z^{12}w^{6}-6652830398677762254200189549054993280z^{11}w^{7}-7988901595095491006415288933717958656z^{10}w^{8}+20623862020729713289196679044991841504z^{9}w^{9}-4603508669395631833386961867060794576z^{8}w^{10}-859429210052536472757318345781651712z^{7}w^{11}-8186306863848773985503789776738354696z^{6}w^{12}-17303895125660293692878041105194211654z^{5}w^{13}+34486411781497343455560463769815890618z^{4}w^{14}-6990910107916430209867639991609015018z^{3}w^{15}-9946410499986751504311657663629676832z^{2}w^{16}+3083472220489336416495545237213620970zw^{17}+497082740331905785733606596690851524w^{18}}{177407626011608614412027978973184xyz^{16}-947930896286022289692835622469632xyz^{15}w-18799425734178683110848551795122176xyz^{14}w^{2}+125992895222373251876227854405394432xyz^{13}w^{3}-134309068059915985915887831932718080xyz^{12}w^{4}-830639443815814370544361549074798592xyz^{11}w^{5}+3174647867791773686282418719662583552xyz^{10}w^{6}-4996741818137682692770956451507437056xyz^{9}w^{7}+4119381108878343074474421667913894656xyz^{8}w^{8}-1137815596776280289694034243485352640xyz^{7}w^{9}-2812774441493063314712480406790893536xyz^{6}w^{10}+7336039793448990090793873541223327872xyz^{5}w^{11}-9823038631165475105441308557484524260xyz^{4}w^{12}+7846010303962817151071475834552501700xyz^{3}w^{13}-3679739098995066162344792280557632527xyz^{2}w^{14}+929569287411771938983490141619609425xyzw^{15}-97006500043197976843015102863145330xyw^{16}-116398735062331718253236008124416xz^{17}-284817563291818589706088698822656xz^{16}w+15973166243140253863479461174697984xz^{15}w^{2}-42060423285074111468794057498722304xz^{14}w^{3}-155610155138802919303204693813576704xz^{13}w^{4}+749914198180492655322263097046130688xz^{12}w^{5}-589926077169548726482687166778340608xz^{11}w^{6}-1785029619083444249565887635639080704xz^{10}w^{7}+4453828451311988689210487238567766272xz^{9}w^{8}-4076914140627897551659343614907544384xz^{8}w^{9}+1259005573571036302100096467978098144xz^{7}w^{10}+2642158882961083133487635389429914912xz^{6}w^{11}-8212597256037279490379167927743163372xz^{5}w^{12}+12292609534921545750704460557436118356xz^{4}w^{13}-10427965841263183225936305582293267203xz^{3}w^{14}+5010471870438887826405360284908854006xz^{2}w^{15}-1260794492700730356798437370890153769xzw^{16}+127327053684911151966444705532131374xw^{17}-132930398609957214812726335700992y^{2}z^{16}+3576614910583279609475180537479168y^{2}z^{15}w-5629941052118523507983488268697600y^{2}z^{14}w^{2}-125972382016103392903381831758073856y^{2}z^{13}w^{3}+645719590327823314660881507777467392y^{2}z^{12}w^{4}-951800089412482396780044765162412032y^{2}z^{11}w^{5}-1176790012644924066083744188716717824y^{2}z^{10}w^{6}+6890283217398029783033975418678511104y^{2}z^{9}w^{7}-13555566505004272290067375926008956928y^{2}z^{8}w^{8}+18274950384767793916085475390078933376y^{2}z^{7}w^{9}-20669451272702373758194751963764791936y^{2}z^{6}w^{10}+19178034459650250707345111514051654832y^{2}z^{5}w^{11}-12042151270258522438237110018248287220y^{2}z^{4}w^{12}+3265978325094116839846151238881314080y^{2}z^{3}w^{13}+1125093763880216775777277757649351705y^{2}z^{2}w^{14}-1090802272817896542318817575530727707y^{2}zw^{15}+234679910774084449270487001824525622y^{2}w^{16}+402908560912955586972118320545792yz^{17}-3595219606531434865824686433501184yz^{16}w-33139106472013471051305379076726784yz^{15}w^{2}+299783053687409467546533582423146496yz^{14}w^{3}-508935137385691609619416990083733504yz^{13}w^{4}-1416078018623265212387877072457596416yz^{12}w^{5}+6807686830675815350560706867714856192yz^{11}w^{6}-10680172462319094916404446553901043200yz^{10}w^{7}+6673483570993284389659025404960186880yz^{9}w^{8}+2272326596598453268921063493703725696yz^{8}w^{9}-10652722391404904763908239901318798240yz^{7}w^{10}+19422141580226374760586013146776093472yz^{6}w^{11}-25355879383088427009963739892539810152yz^{5}w^{12}+19944949903335629221660041644842425942yz^{4}w^{13}-6693968220516320695235325463958303839yz^{3}w^{14}-1641091823583455668061983660765287083yz^{2}w^{15}+2034454785390240622581832531179706536yzw^{16}-469704935557924371397057885463971420yw^{17}-204839540869886970625555256115200z^{18}+331119697227592572663700074659840z^{17}w+23093251965168703722441121772363776z^{16}w^{2}-91545895414963491132807069953781760z^{15}w^{3}-116665613164620632210714501487657984z^{14}w^{4}+1095487736408092212091669020532359680z^{13}w^{5}-1551208817266288581263911123505222144z^{12}w^{6}-1180554907860983263191699907927849984z^{11}w^{7}+5616474862646197289034449093703758848z^{10}w^{8}-6979831383261418515673138466800125056z^{9}w^{9}+5590189910337744418272613570514416032z^{8}w^{10}-4033030483129521273101534954349378768z^{7}w^{11}+402312086603636992812034800252467140z^{6}w^{12}+4741957751946659244596220135008018246z^{5}w^{13}-5817444934202592011119370919102928850z^{4}w^{14}+2284661523031681816106365657147963318z^{3}w^{15}+630859694231148278473601443927952748z^{2}w^{16}-810007325201401141654681341469642990zw^{17}+195165837359375305376483839846696100w^{18}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 13.84.2.c.2 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 13w$
$\displaystyle Z$ $=$ $\displaystyle y$

Equation of the image curve:

$0$ $=$ $ 17X^{5}-2X^{4}Y-X^{3}Y^{2}+67X^{4}Z-19X^{3}YZ+X^{2}Y^{2}Z+211X^{3}Z^{2}-36X^{2}YZ^{2}+4XY^{2}Z^{2}+220X^{2}Z^{3}-57XYZ^{3}+Y^{2}Z^{3}+206XZ^{4}-18YZ^{4}+79Z^{5} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 13.84.2.c.2 :

$\displaystyle X$ $=$ $\displaystyle xy-\frac{1}{3}y^{2}$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}x^{6}+\frac{22}{9}x^{5}y+\frac{13}{3}x^{5}w+\frac{106}{27}x^{4}y^{2}-\frac{65}{9}x^{4}yw+\frac{167}{27}x^{3}y^{3}-\frac{377}{27}x^{3}y^{2}w-\frac{76}{27}x^{2}y^{4}+\frac{182}{27}x^{2}y^{3}w-\frac{25}{27}xy^{5}+\frac{26}{27}xy^{4}w+\frac{1}{3}y^{6}-\frac{13}{27}y^{5}w$
$\displaystyle Z$ $=$ $\displaystyle -x^{2}-\frac{2}{3}xy+\frac{1}{3}y^{2}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
13.56.0-13.a.2.2 $13$ $3$ $3$ $0$ $0$ 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
13.2184.50-13.c.1.1 $13$ $13$ $13$ $50$ $9$ $2^{6}\cdot3^{6}\cdot6^{3}$
26.336.9-26.e.1.1 $26$ $2$ $2$ $9$ $0$ $1^{5}\cdot2$
26.336.9-26.i.1.1 $26$ $2$ $2$ $9$ $0$ $1^{5}\cdot2$
26.504.10-26.c.2.2 $26$ $3$ $3$ $10$ $0$ $1^{4}\cdot2^{2}$
39.504.16-39.e.1.3 $39$ $3$ $3$ $16$ $2$ $1^{2}\cdot2^{2}\cdot4^{2}$
39.672.17-39.e.1.3 $39$ $4$ $4$ $17$ $2$ $1^{3}\cdot2^{6}$
52.336.9-52.p.1.3 $52$ $2$ $2$ $9$ $0$ $1^{5}\cdot2$
52.336.9-52.bf.1.2 $52$ $2$ $2$ $9$ $0$ $1^{5}\cdot2$
52.672.23-52.cg.1.1 $52$ $4$ $4$ $23$ $7$ $1^{3}\cdot2^{3}\cdot4\cdot8$
65.840.30-65.e.1.3 $65$ $5$ $5$ $30$ $11$ $1^{2}\cdot2^{3}\cdot5^{2}\cdot10$
65.1008.31-65.i.1.2 $65$ $6$ $6$ $31$ $5$ $1\cdot2^{7}\cdot6\cdot8$
65.1680.59-65.r.2.3 $65$ $10$ $10$ $59$ $21$ $1^{3}\cdot2^{10}\cdot5^{2}\cdot6\cdot8\cdot10$
78.336.9-78.h.1.1 $78$ $2$ $2$ $9$ $?$ not computed
78.336.9-78.z.1.1 $78$ $2$ $2$ $9$ $?$ not computed
104.336.9-104.bz.1.3 $104$ $2$ $2$ $9$ $?$ not computed
104.336.9-104.cf.1.3 $104$ $2$ $2$ $9$ $?$ not computed
104.336.9-104.eh.1.3 $104$ $2$ $2$ $9$ $?$ not computed
104.336.9-104.en.1.3 $104$ $2$ $2$ $9$ $?$ not computed
130.336.9-130.h.1.1 $130$ $2$ $2$ $9$ $?$ not computed
130.336.9-130.x.1.1 $130$ $2$ $2$ $9$ $?$ not computed
156.336.9-156.y.1.1 $156$ $2$ $2$ $9$ $?$ not computed
156.336.9-156.dq.1.1 $156$ $2$ $2$ $9$ $?$ not computed
169.2184.50-169.c.2.1 $169$ $13$ $13$ $50$ $?$ not computed
182.336.9-182.bj.2.3 $182$ $2$ $2$ $9$ $?$ not computed
182.336.9-182.dg.2.3 $182$ $2$ $2$ $9$ $?$ not computed
260.336.9-260.y.1.3 $260$ $2$ $2$ $9$ $?$ not computed
260.336.9-260.do.1.3 $260$ $2$ $2$ $9$ $?$ not computed
286.336.9-286.h.1.1 $286$ $2$ $2$ $9$ $?$ not computed
286.336.9-286.x.1.1 $286$ $2$ $2$ $9$ $?$ not computed
312.336.9-312.dj.1.1 $312$ $2$ $2$ $9$ $?$ not computed
312.336.9-312.dp.1.1 $312$ $2$ $2$ $9$ $?$ not computed
312.336.9-312.nf.1.1 $312$ $2$ $2$ $9$ $?$ not computed
312.336.9-312.nl.1.1 $312$ $2$ $2$ $9$ $?$ not computed