Modular curve 7.42.1.b.1

• Label: 7.42.1.b.1
• Level: $7$
• Index: $42$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $6$
• $\Q$-cusps: $0$

Invariants

Level:	$7$		$\SL_2$-level:	$7$		Newform level:	$49$
Index:	$42$		$\PSL_2$-index:	$42$
Genus:	$1 = 1 + \frac{ 42 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (none of which are rational)		Cusp widths	$7^{6}$		Cusp orbits	$3^{2}$
Elliptic points:	$2$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	yes $\quad(D =$ $-3$)

Other labels

Cummins and Pauli (CP) label:	7A1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	7.42.1.1
Sutherland (S) label:	7Nn.1.3

Level structure

$\GL_2(\Z/7\Z)$-generators:	$\begin{bmatrix}4&6\\0&3\end{bmatrix}$, $\begin{bmatrix}6&0\\1&1\end{bmatrix}$
$\GL_2(\Z/7\Z)$-subgroup:	$C_3\times D_8$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 7-isogeny field degree:	$4$
Cyclic 7-torsion field degree:	$24$
Full 7-torsion field degree:	$48$

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} - 1822x + 30393 $

Rational points

This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve	CM	$j$-invariant		$j$-height
27.a3	$-3$	$0$		$0.000$

Maps to other modular curves

$j$-invariant map of degree 42 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 7^5\,\frac{z(411136x^{2}y^{25}-2406991168x^{2}y^{24}z-7251914976888x^{2}y^{23}z^{2}+34922693732948849x^{2}y^{22}z^{3}-3806820278770219622x^{2}y^{21}z^{4}-84921906868311447672528x^{2}y^{20}z^{5}+57979578174039464025053404x^{2}y^{19}z^{6}+60910437428964030272217968754x^{2}y^{18}z^{7}-68238068181521357976597991017956x^{2}y^{17}z^{8}-10230869899503774478472268660901977x^{2}y^{16}z^{9}+31684901503968593572766565708202665080x^{2}y^{15}z^{10}-4720044937861059602378091635560937177801x^{2}y^{14}z^{11}-7134756566542950850234373752033576758421517x^{2}y^{13}z^{12}+2381723163340358252985709852234430498916121740x^{2}y^{12}z^{13}+775050716754878779602908391092071245046736023529x^{2}y^{11}z^{14}-445696086307097891517466123013275165915179441034148x^{2}y^{10}z^{15}-22701715128958142696397021284031004724216668633020785x^{2}y^{9}z^{16}+43270178951683791311132896292449353657859483457085522125x^{2}y^{8}z^{17}-3398820006321074146701555919032659850574811778616629059826x^{2}y^{7}z^{18}-2219515948792152277480104612374805190629387893538251546124744x^{2}y^{6}z^{19}+381277487400774248129194882603320186689146342814462533198085684x^{2}y^{5}z^{20}+49414054065600997502972303743671502939310056188854364394214705488x^{2}y^{4}z^{21}-15269978527106173312778572681889996842735722993717513865755656689776x^{2}y^{3}z^{22}+110460765647506212229826675128635605185058786039881302971586019563424x^{2}y^{2}z^{23}+224339369633398491623944222257726654753610942041311831677889374667561792x^{2}yz^{24}-16064550339065263697162554961766897833820863219960519294358737622953424384x^{2}z^{25}-5120xy^{26}-160759168xy^{25}z+790459965584xy^{24}z^{2}+863457689489702xy^{23}z^{3}-5646944116979674967xy^{22}z^{4}+2223293578767946747471xy^{21}z^{5}+9048246207760713999830946xy^{20}z^{6}-7535060019878859907706026925xy^{19}z^{7}-4524712959299480249406431968022xy^{18}z^{8}+6400236017331699047582090177577091xy^{17}z^{9}+178501891436463751501751993001913857xy^{16}z^{10}-2413434689667322510882811991956383947615xy^{15}z^{11}+509951577190065648495777207264977887203762xy^{14}z^{12}+459121290665642993863987009889295008489353927xy^{13}z^{13}-180063213457575152255471139125251103216998985675xy^{12}z^{14}-41372572892284758828689655640012636836235440687337xy^{11}z^{15}+28801477505496826320702320937765099280410584621299079xy^{10}z^{16}+459483210745205585933869233053738523574364984281480589xy^{9}z^{17}-2505849174540542530075971999100105782665152841401262373321xy^{8}z^{18}+244738031305685427969878446130937608918082815099654827142148xy^{7}z^{19}+117151168703831886514398123884245541633100635850307472706483930xy^{6}z^{20}-22074527947383754395485042906552443109479853307771045284169353544xy^{5}z^{21}-2337350105081188386208007548830077752735478656007430700397113793832xy^{4}z^{22}+803559148394718321342059102756769447902473166789333611226229791904528xy^{3}z^{23}-10477992718397582817956536281524729114228161980210556364314450725867744xy^{2}z^{24}-11057416438021896131201530733298007040877995363890962133031089764724014592xyz^{25}+802114197304056128496314175854103695525269211058560774330605274426380034048xz^{26}-512y^{27}+4799936y^{26}z+41514471912y^{25}z^{2}-180651909799243y^{24}z^{3}-68166322277725445y^{23}z^{4}+718664570772655999130y^{22}z^{5}-369903929281017052702809y^{21}z^{6}-758191009963110263981438038y^{20}z^{7}+688893543553301346031582306740y^{19}z^{8}+242396723603647191420930945696528y^{18}z^{9}-420764998948553230955453087881804504y^{17}z^{10}+23711595072978697290234923042318921588y^{16}z^{11}+120310418869573881521786465723612920381459y^{15}z^{12}-30986980146384643852005518708571506784738450y^{14}z^{13}-17359098573842473099621689823233132458492937343y^{13}z^{14}+7724828884904487209161477026827834603547642859230y^{12}z^{15}+1071550742616983821275640628953562708948065202122178y^{11}z^{16}-954931858541475315561160786705798822694128421018783941y^{10}z^{17}+23301819104326622169255516719015476559316327537349709319y^{9}z^{18}+65125803901510449835147778761051743210830113168624145501330y^{8}z^{19}-7889014615350849076409443966004721675392419290936692099462103y^{7}z^{20}-2341483466932479559146755415188459292646948497461413896924408590y^{6}z^{21}+494461727083419603098204641445617603147966597700580754212302529932y^{5}z^{22}+31829975097871089719644127843375692227696134817640698043049526466432y^{4}z^{23}-13589399333055659510049161970859908721715305435068391067966943376962464y^{3}z^{24}+333592288265704287500024075611362244178855257353384254237631311950600736y^{2}z^{25}+141175059503627131996091910135308762429418034154740433892607186360306408320yz^{26}-9978396123922642766479307412031554207911308694346135642572155652963949091968z^{27})}{763x^{2}y^{26}-4471285x^{2}y^{25}z-15452165985x^{2}y^{24}z^{2}+44442829028921x^{2}y^{23}z^{3}+43405002583888653x^{2}y^{22}z^{4}-93976480553372374646x^{2}y^{21}z^{5}-44007205852724941077958x^{2}y^{20}z^{6}+79731311098926517964628205x^{2}y^{19}z^{7}+22919780268759252449075716617x^{2}y^{18}z^{8}-35309662087892059722236670671221x^{2}y^{17}z^{9}-7296276310286697536603736752811185x^{2}y^{16}z^{10}+9334942000107147853653760687415172170x^{2}y^{15}z^{11}+1556511302917651498565920369418021637012x^{2}y^{14}z^{12}-1585990382386738316711631968432340840323934x^{2}y^{13}z^{13}-232731277063156223361691279642160541277251448x^{2}y^{12}z^{14}+179997461943811999949929051822293797738210105144x^{2}y^{11}z^{15}+24752411359986126562866055488285982541531133682752x^{2}y^{10}z^{16}-13832140603237564917094500237189452360867721241849664x^{2}y^{9}z^{17}-1860699429893439239567866080362418171833250579442157440x^{2}y^{8}z^{18}+712445446462621572391945846079933060780019152691278856960x^{2}y^{7}z^{19}+96366019751123909770586668821997076593872806423371693476864x^{2}y^{6}z^{20}-23601792322031983856681166868405733630739483274499109757487616x^{2}y^{5}z^{21}-3263264212159822312547836015930047687790325186851248355700745216x^{2}y^{4}z^{22}+455224489327399841008615448924373346591926623541263478694415175680x^{2}y^{3}z^{23}+64919108647749479127431188816511015050649972429618177649438049501184x^{2}y^{2}z^{24}-3890233158700425238453101663375568487596929717427228507564667009335296x^{2}yz^{25}-574607273649421261534079675672926517120525662377896195872819722165944320x^{2}z^{26}-14xy^{27}-292103xy^{26}z+1267217341xy^{25}z^{2}+2505585911658xy^{24}z^{3}-6797165482100131xy^{23}z^{4}-4766722379530179945xy^{22}z^{5}+10465868917463847328544xy^{21}z^{6}+3835714399603265959571029xy^{20}z^{7}-7275752417034360489546915431xy^{19}z^{8}-1716959645377827912836502533536xy^{18}z^{9}+2805530506573567351935726225270981xy^{17}z^{10}+492772659736569818115063200679577905xy^{16}z^{11}-669408347542104531998444712641025192354xy^{15}z^{12}-97731909235638243013838388890951243311906xy^{14}z^{13}+105049322148198473130182914937623568756898172xy^{13}z^{14}+13854065106280891510410974690706334974602546244xy^{12}z^{15}-11188493545547691466922879308981432375261949080424xy^{11}z^{16}-1413870412434131610788819535239656890222410240388752xy^{10}z^{17}+816135740079265219378148320895153746551413539044589024xy^{9}z^{18}+102736001994518820154400244351333674996302543354057599360xy^{8}z^{19}-40241955561418602338653928606577261660453415138351118834944xy^{7}z^{20}-5167076367319076764994231798681667740442698830189554413440512xy^{6}z^{21}+1284540837939829516448234501980990280968258091371926585172690944xy^{5}z^{22}+170469550668733985099749956269786165461312638589943342177644123136xy^{4}z^{23}-23994556354229367503784408222758119686499643032759735225130001006592xy^{3}z^{24}-3312158656911829387025964423969441345326711791810924297167821509840896xy^{2}z^{25}+199392906466591892517971083321854804122008088678587957453485871435612160xyz^{26}+28690541741935600835252558618663409376034402553190168956136192211100729344xz^{27}-y^{28}+10738y^{27}z+78246070y^{26}z^{2}-251826408084y^{25}z^{3}-361000176856669y^{24}z^{4}+792856209277125998y^{23}z^{5}+489091546608395310896y^{22}z^{6}-864967846907876294037462y^{21}z^{7}-311238313669192658805673211y^{20}z^{8}+461575713936802262332255999054y^{19}z^{9}+115392306596266660384911290285459y^{18}z^{10}-142489328749759123039762119573144956y^{17}z^{11}-27905574390238715053642766538627136560y^{16}z^{12}+27887808971237513009130126626997121691176y^{15}z^{13}+4671321730161132063362267032215167473717465y^{14}z^{14}-3641873313609451107448148213518379697405541072y^{13}z^{15}-556256009562080971578401185324131103927542612612y^{12}z^{16}+325390929770658607592439982408328921462978224702640y^{11}z^{17}+47408708577752815386959116204933221118114580282126480y^{10}z^{18}-19979540583503841185731761929243047076530377074312792800y^{9}z^{19}-2862184060469130453722493933912175083721213459531315256832y^{8}z^{20}+828937324566097143183994005360027375595222930600781994190976y^{7}z^{21}+119034037953989645273704220413623524675286271070698007712748544y^{6}z^{22}-22175996055014036676629592412261053399362889146415754238854011904y^{5}z^{23}-3228790905913399357815984624359525955132139290121148024289888801792y^{4}z^{24}+344468765177680488307639839993162228755498509927524712899746271692800y^{3}z^{25}+51157891766937626134994921142649528107306974248369553955234756575002624y^{2}z^{26}-2351007501090007814780277263916122398268103765453592486074360502345703424yz^{27}-356913755514103152055331334175857394787242865006682219945959271121955373056z^{28}}$

Modular covers

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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.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{ns}}^+(7)$	$7$	$2$	$2$	$0$	$0$	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
7.84.1.a.1	$7$	$2$	$2$	$1$	$0$	dimension zero
14.84.5.b.1	$14$	$2$	$2$	$5$	$0$	$1^{2}\cdot2$
14.84.5.d.1	$14$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
14.84.5.e.1	$14$	$2$	$2$	$5$	$1$	$1^{2}\cdot2$
14.126.5.b.1	$14$	$3$	$3$	$5$	$0$	$1^{2}\cdot2$
21.84.1.a.1	$21$	$2$	$2$	$1$	$0$	dimension zero
21.126.7.c.1	$21$	$3$	$3$	$7$	$2$	$1^{2}\cdot2^{2}$
21.168.9.c.1	$21$	$4$	$4$	$9$	$2$	$1^{4}\cdot2^{2}$
28.84.1.a.1	$28$	$2$	$2$	$1$	$0$	dimension zero
28.84.5.e.1	$28$	$2$	$2$	$5$	$0$	$1^{2}\cdot2$
28.84.5.k.1	$28$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
28.84.5.m.1	$28$	$2$	$2$	$5$	$1$	$1^{2}\cdot2$
28.168.11.g.1	$28$	$4$	$4$	$11$	$7$	$1^{6}\cdot2^{2}$
35.84.1.a.1	$35$	$2$	$2$	$1$	$0$	dimension zero
35.210.15.c.1	$35$	$5$	$5$	$15$	$10$	$2^{2}\cdot3^{2}\cdot4$
35.252.15.c.1	$35$	$6$	$6$	$15$	$3$	$1^{4}\cdot2^{5}$
35.420.29.e.1	$35$	$10$	$10$	$29$	$19$	$1^{4}\cdot2^{7}\cdot3^{2}\cdot4$
42.84.5.d.1	$42$	$2$	$2$	$5$	$4$	$1^{2}\cdot2$
42.84.5.k.1	$42$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
42.84.5.m.1	$42$	$2$	$2$	$5$	$3$	$1^{2}\cdot2$
49.294.19.e.1	$49$	$7$	$7$	$19$	$18$	$9^{2}$
49.2058.151.b.1	$49$	$49$	$49$	$151$	$102$	$3^{2}\cdot6^{2}\cdot9^{2}\cdot18\cdot24^{2}\cdot48$
56.84.1.a.1	$56$	$2$	$2$	$1$	$0$	dimension zero
56.84.1.b.1	$56$	$2$	$2$	$1$	$0$	dimension zero
56.84.5.n.1	$56$	$2$	$2$	$5$	$0$	$1^{2}\cdot2$
56.84.5.t.1	$56$	$2$	$2$	$5$	$4$	$1^{2}\cdot2$
56.84.5.bo.1	$56$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
56.84.5.br.1	$56$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
56.84.5.bw.1	$56$	$2$	$2$	$5$	$1$	$1^{2}\cdot2$
56.84.5.bx.1	$56$	$2$	$2$	$5$	$3$	$1^{2}\cdot2$
63.1134.85.c.1	$63$	$27$	$27$	$85$	$54$	$1^{11}\cdot2^{2}\cdot3^{7}\cdot4^{5}\cdot5^{2}\cdot6\cdot12$
70.84.5.d.1	$70$	$2$	$2$	$5$	$0$	$1^{2}\cdot2$
70.84.5.k.1	$70$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
70.84.5.m.1	$70$	$2$	$2$	$5$	$1$	$1^{2}\cdot2$
77.84.1.a.1	$77$	$2$	$2$	$1$	$?$	dimension zero
84.84.1.a.1	$84$	$2$	$2$	$1$	$?$	dimension zero
84.84.5.k.1	$84$	$2$	$2$	$5$	$?$	not computed
84.84.5.bf.1	$84$	$2$	$2$	$5$	$?$	not computed
84.84.5.bk.1	$84$	$2$	$2$	$5$	$?$	not computed
91.84.1.a.1	$91$	$2$	$2$	$1$	$?$	dimension zero
105.84.1.a.1	$105$	$2$	$2$	$1$	$?$	dimension zero
119.84.1.a.1	$119$	$2$	$2$	$1$	$?$	dimension zero
133.84.1.a.1	$133$	$2$	$2$	$1$	$?$	dimension zero
140.84.1.a.1	$140$	$2$	$2$	$1$	$?$	dimension zero
140.84.5.k.1	$140$	$2$	$2$	$5$	$?$	not computed
140.84.5.bf.1	$140$	$2$	$2$	$5$	$?$	not computed
140.84.5.bk.1	$140$	$2$	$2$	$5$	$?$	not computed
154.84.5.g.1	$154$	$2$	$2$	$5$	$?$	not computed
154.84.5.h.1	$154$	$2$	$2$	$5$	$?$	not computed
154.84.5.i.1	$154$	$2$	$2$	$5$	$?$	not computed
161.84.1.a.1	$161$	$2$	$2$	$1$	$?$	dimension zero
168.84.1.a.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.84.1.b.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.84.5.bl.1	$168$	$2$	$2$	$5$	$?$	not computed
168.84.5.br.1	$168$	$2$	$2$	$5$	$?$	not computed
168.84.5.eu.1	$168$	$2$	$2$	$5$	$?$	not computed
168.84.5.ex.1	$168$	$2$	$2$	$5$	$?$	not computed
168.84.5.fo.1	$168$	$2$	$2$	$5$	$?$	not computed
168.84.5.fp.1	$168$	$2$	$2$	$5$	$?$	not computed
182.84.5.bb.1	$182$	$2$	$2$	$5$	$?$	not computed
182.84.5.bc.1	$182$	$2$	$2$	$5$	$?$	not computed
182.84.5.bd.1	$182$	$2$	$2$	$5$	$?$	not computed
203.84.1.a.1	$203$	$2$	$2$	$1$	$?$	dimension zero
210.84.5.k.1	$210$	$2$	$2$	$5$	$?$	not computed
210.84.5.bf.1	$210$	$2$	$2$	$5$	$?$	not computed
210.84.5.bk.1	$210$	$2$	$2$	$5$	$?$	not computed
217.84.1.a.1	$217$	$2$	$2$	$1$	$?$	dimension zero
231.84.1.a.1	$231$	$2$	$2$	$1$	$?$	dimension zero
238.84.5.g.1	$238$	$2$	$2$	$5$	$?$	not computed
238.84.5.h.1	$238$	$2$	$2$	$5$	$?$	not computed
238.84.5.i.1	$238$	$2$	$2$	$5$	$?$	not computed
259.84.1.a.1	$259$	$2$	$2$	$1$	$?$	dimension zero
266.84.5.g.1	$266$	$2$	$2$	$5$	$?$	not computed
266.84.5.h.1	$266$	$2$	$2$	$5$	$?$	not computed
266.84.5.i.1	$266$	$2$	$2$	$5$	$?$	not computed
273.84.1.a.1	$273$	$2$	$2$	$1$	$?$	dimension zero
280.84.1.a.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.84.1.b.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.84.5.bl.1	$280$	$2$	$2$	$5$	$?$	not computed
280.84.5.br.1	$280$	$2$	$2$	$5$	$?$	not computed
280.84.5.eu.1	$280$	$2$	$2$	$5$	$?$	not computed
280.84.5.ex.1	$280$	$2$	$2$	$5$	$?$	not computed
280.84.5.fo.1	$280$	$2$	$2$	$5$	$?$	not computed
280.84.5.fp.1	$280$	$2$	$2$	$5$	$?$	not computed
287.84.1.a.1	$287$	$2$	$2$	$1$	$?$	dimension zero
301.84.1.a.1	$301$	$2$	$2$	$1$	$?$	dimension zero
308.84.1.a.1	$308$	$2$	$2$	$1$	$?$	dimension zero
308.84.5.t.1	$308$	$2$	$2$	$5$	$?$	not computed
308.84.5.w.1	$308$	$2$	$2$	$5$	$?$	not computed
308.84.5.y.1	$308$	$2$	$2$	$5$	$?$	not computed
322.84.5.g.1	$322$	$2$	$2$	$5$	$?$	not computed
322.84.5.h.1	$322$	$2$	$2$	$5$	$?$	not computed
322.84.5.i.1	$322$	$2$	$2$	$5$	$?$	not computed
329.84.1.a.1	$329$	$2$	$2$	$1$	$?$	dimension zero