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Properties

Label 40.48.1.ip.2
Level $40$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $800$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ 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: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.218

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}9&22\\21&5\end{bmatrix}$, $\begin{bmatrix}29&10\\23&33\end{bmatrix}$, $\begin{bmatrix}33&6\\17&25\end{bmatrix}$, $\begin{bmatrix}39&2\\23&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.ip.2.1, 40.96.1-40.ip.2.2, 40.96.1-40.ip.2.3, 40.96.1-40.ip.2.4, 40.96.1-40.ip.2.5, 40.96.1-40.ip.2.6, 40.96.1-40.ip.2.7, 40.96.1-40.ip.2.8, 80.96.1-40.ip.2.1, 80.96.1-40.ip.2.2, 80.96.1-40.ip.2.3, 80.96.1-40.ip.2.4, 80.96.1-40.ip.2.5, 80.96.1-40.ip.2.6, 80.96.1-40.ip.2.7, 80.96.1-40.ip.2.8, 120.96.1-40.ip.2.1, 120.96.1-40.ip.2.2, 120.96.1-40.ip.2.3, 120.96.1-40.ip.2.4, 120.96.1-40.ip.2.5, 120.96.1-40.ip.2.6, 120.96.1-40.ip.2.7, 120.96.1-40.ip.2.8, 240.96.1-40.ip.2.1, 240.96.1-40.ip.2.2, 240.96.1-40.ip.2.3, 240.96.1-40.ip.2.4, 240.96.1-40.ip.2.5, 240.96.1-40.ip.2.6, 240.96.1-40.ip.2.7, 240.96.1-40.ip.2.8, 280.96.1-40.ip.2.1, 280.96.1-40.ip.2.2, 280.96.1-40.ip.2.3, 280.96.1-40.ip.2.4, 280.96.1-40.ip.2.5, 280.96.1-40.ip.2.6, 280.96.1-40.ip.2.7, 280.96.1-40.ip.2.8
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
Full 40-torsion field degree: $15360$

Jacobian

Conductor: $2^{5}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 800.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 275x + 1750 $
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Rational points

This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^4\,\frac{10081000x^{2}y^{30}-28482810000x^{2}y^{29}z+33751234962500x^{2}y^{28}z^{2}-4278914653000000x^{2}y^{27}z^{3}-14640511278353750000x^{2}y^{26}z^{4}+4994638840607843750000x^{2}y^{25}z^{5}+1426400106280455527343750x^{2}y^{24}z^{6}-463541492514816000000000000x^{2}y^{23}z^{7}-89216792279401842734375000000x^{2}y^{22}z^{8}+14007264067985602366210937500000x^{2}y^{21}z^{9}+3624155735602978521041107177734375x^{2}y^{20}z^{10}-46514975646553486124389648437500000x^{2}y^{19}z^{11}-75889635682256764110838546752929687500x^{2}y^{18}z^{12}-5551139437750922919490550994873046875000x^{2}y^{17}z^{13}+650692391821077089735599396228790283203125x^{2}y^{16}z^{14}+117482604856949005748142578125000000000000000x^{2}y^{15}z^{15}+1512139498257823249972267534732818603515625000x^{2}y^{14}z^{16}-979380718383044496144133777499198913574218750000x^{2}y^{13}z^{17}-74502142572745583487966941426098346710205078125000x^{2}y^{12}z^{18}+2569896145330913673233464961707592010498046875000000x^{2}y^{11}z^{19}+602437748752603278212961264829784631729125976562500000x^{2}y^{10}z^{20}+15890841755516290465792778425476513803005218505859375000x^{2}y^{9}z^{21}-2126440084171925749760000975353331887163221836090087890625x^{2}y^{8}z^{22}-153071520471460161257472014937847852706909179687500000000000x^{2}y^{7}z^{23}+2001595193278602118430720119021300924941897392272949218750000x^{2}y^{6}z^{24}+530337749734674093047807999518807948334142565727233886718750000x^{2}y^{5}z^{25}+8778820410506786101002240001267820452994783408939838409423828125x^{2}y^{4}z^{26}-880443521830952942698496000002285603841301053762435913085937500000x^{2}y^{3}z^{27}-27734667689218880768573439999997238669493526685982942581176757812500x^{2}y^{2}z^{28}+583203410662266515750911999999997962959241704083979129791259765625000x^{2}yz^{29}+24035680807957675512954880000000000698179292157874442636966705322265625x^{2}z^{30}+358360xy^{31}-692311000xy^{30}z+2577893870000xy^{29}z^{2}-3476982996550000xy^{28}z^{3}+888685234637500000xy^{27}z^{4}+760260903060107031250xy^{26}z^{5}-268503327743381250000000xy^{25}z^{6}-61998925914177214648437500xy^{24}z^{7}+17525247240862634863281250000xy^{23}z^{8}+3507900042857122767041015625000xy^{22}z^{9}-382544694568129504487304687500000xy^{21}z^{10}-120439686963770501952069091796875000xy^{20}z^{11}-1433010443984342745979766845703125000xy^{19}z^{12}+2081819978510176878183441257476806640625xy^{18}z^{13}+189474626302653161188742332458496093750000xy^{17}z^{14}-13405289691583786616836438727378845214843750xy^{16}z^{15}-3148729866393617380349293076992034912109375000xy^{15}z^{16}-82110546167551182971024852180480957031250000000xy^{14}z^{17}+22212114595723629008346670344471931457519531250000xy^{13}z^{18}+1980946999672774538084209511482715606689453125000000xy^{12}z^{19}-39662878150710284779329648762941360473632812500000000xy^{11}z^{20}-13836231668020952852072081837987690232694149017333984375xy^{10}z^{21}-447260994119338830069752260997239500284194946289062500000xy^{9}z^{22}+43678510217590121221324793059989233734086155891418457031250xy^{8}z^{23}+3492085044952767764889599835391946253366768360137939453125000xy^{7}z^{24}-30143764476696044398182398764249026135075837373733520507812500xy^{6}z^{25}-11160149912297108248985600004899433552054688334465026855468750000xy^{5}z^{26}-199228437225689683538739199987212682754034176468849182128906250000xy^{4}z^{27}+17561113685390438013337599999977062479956657625734806060791015625000xy^{3}z^{28}+560060940919249825274265600000027641232236419455148279666900634765625xy^{2}z^{29}-11163758783119562710712320000000020370407582959160208702087402343750000xyz^{30}-460094261834632934929203199999999993018207078421255573630332946777343750xz^{31}+4913y^{32}+107960800y^{31}z-159131304000y^{30}z^{2}-89601786000000y^{29}z^{3}+248309127422875000y^{28}z^{4}-77955872663850000000y^{27}z^{5}-32399296452973906250000y^{26}z^{6}+11358655590931633593750000y^{25}z^{7}+2240904541786520272460937500y^{24}z^{8}-483081997621421481640625000000y^{23}z^{9}-107488868136910411291992187500000y^{22}z^{10}+5309160144348983883306884765625000y^{21}z^{11}+2816872798560696951184303283691406250y^{20}z^{12}+137234411360185903534478759765625000000y^{19}z^{13}-33577676802922789429479728698730468750000y^{18}z^{14}-4468322741901841583429253673553466796875000y^{17}z^{15}+63045833624122421907403581857681274414062500y^{16}z^{16}+49271505411082257890511215066909790039062500000y^{15}z^{17}+2664210007723291550136213383436203002929687500000y^{14}z^{18}-219142822940318157510619894164800643920898437500000y^{13}z^{19}-30488049429955345384379377955630421638488769531250000y^{12}z^{20}-259226985766806408399529059383273124694824218750000000y^{11}z^{21}+147346572537801023160345933268868364393711090087890625000y^{10}z^{22}+7519450292337058093465462414533435367047786712646484375000y^{9}z^{23}-301098050072386100789247771425617655040696263313293457031250y^{8}z^{24}-36659371063046851002367997693812206853181123733520507812500000y^{7}z^{25}-164675034794702235762688019913657481083646416664123535156250000y^{6}z^{26}+879466429804958818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Modular covers

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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
8.24.0.bl.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.el.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.ea.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
40.240.17.yl.2 $40$ $5$ $5$ $17$ $2$ $1^{6}\cdot2^{5}$
40.288.17.cfn.1 $40$ $6$ $6$ $17$ $3$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.dxn.2 $40$ $10$ $10$ $33$ $3$ $1^{12}\cdot2^{6}\cdot4^{2}$
80.96.5.hm.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ho.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ji.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.jk.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.so.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.sq.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.uk.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.um.2 $80$ $2$ $2$ $5$ $?$ not computed
120.144.9.beyf.2 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.dop.2 $120$ $4$ $4$ $9$ $?$ not computed
240.96.5.cfo.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cfq.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cge.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cgg.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dqy.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dra.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dro.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.drq.2 $240$ $2$ $2$ $5$ $?$ not computed