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Properties

Label 70.48.2.c.1
Level $70$
Index $48$
Genus $2$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $70$ $\SL_2$-level: $14$ Newform level: $350$
Index: $48$ $\PSL_2$-index:$48$
Genus: $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $2^{3}\cdot14^{3}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ 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: 14E2
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 70.48.2.7

Level structure

$\GL_2(\Z/70\Z)$-generators: $\begin{bmatrix}25&22\\68&29\end{bmatrix}$, $\begin{bmatrix}49&38\\10&21\end{bmatrix}$, $\begin{bmatrix}67&45\\64&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 70.96.2-70.c.1.1, 70.96.2-70.c.1.2, 70.96.2-70.c.1.3, 70.96.2-70.c.1.4, 140.96.2-70.c.1.1, 140.96.2-70.c.1.2, 140.96.2-70.c.1.3, 140.96.2-70.c.1.4, 140.96.2-70.c.1.5, 140.96.2-70.c.1.6, 140.96.2-70.c.1.7, 140.96.2-70.c.1.8, 140.96.2-70.c.1.9, 140.96.2-70.c.1.10, 140.96.2-70.c.1.11, 140.96.2-70.c.1.12, 210.96.2-70.c.1.1, 210.96.2-70.c.1.2, 210.96.2-70.c.1.3, 210.96.2-70.c.1.4, 280.96.2-70.c.1.1, 280.96.2-70.c.1.2, 280.96.2-70.c.1.3, 280.96.2-70.c.1.4, 280.96.2-70.c.1.5, 280.96.2-70.c.1.6, 280.96.2-70.c.1.7, 280.96.2-70.c.1.8, 280.96.2-70.c.1.9, 280.96.2-70.c.1.10, 280.96.2-70.c.1.11, 280.96.2-70.c.1.12, 280.96.2-70.c.1.13, 280.96.2-70.c.1.14, 280.96.2-70.c.1.15, 280.96.2-70.c.1.16
Cyclic 70-isogeny field degree: $6$
Cyclic 70-torsion field degree: $144$
Full 70-torsion field degree: $120960$

Jacobian

Conductor: $2^{2}\cdot5^{2}\cdot7^{2}$
Simple: no
Squarefree: yes
Decomposition: $1^{2}$
Newforms: 14.2.a.a, 350.2.a.f

Models

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

$ 0 $ $=$ $ 5 x^{2} w + 2 x y w - 5 x w^{2} + y^{2} w - y z w $
$=$ $5 x^{2} y + 2 x y^{2} - 5 x y w + y^{3} - y^{2} z$
$=$ $5 x^{3} + 2 x^{2} y - 5 x^{2} w + x y^{2} - x y z$
$=$ $5 x^{2} z + 2 x y z - 5 x z w + y^{2} z - y z^{2}$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 4 x^{3} y + 10 x^{3} z + 5 x^{2} y^{2} - 25 x^{2} y z + 35 x^{2} z^{2} - 2 x y^{3} + \cdots - 25 z^{4} $
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Weierstrass model Weierstrass model

$ y^{2} + \left(x^{2} + x + 1\right) y $ $=$ $ x^{6} + 3x^{5} + 16x^{4} + 27x^{3} + 58x^{2} + 45x + 65 $
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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.

Embedded model
$(0:0:1:0)$, $(0:0:0:1)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{5}y$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle -x^{2}-\frac{3}{5}xy+xw$
$\displaystyle Y$ $=$ $\displaystyle -\frac{1}{2}x^{6}-\frac{3}{5}x^{5}y+x^{5}w-\frac{3}{10}x^{4}y^{2}+\frac{7}{10}x^{4}yw-\frac{1}{2}x^{4}w^{2}+\frac{9}{125}x^{3}y^{3}-\frac{1}{10}x^{3}y^{2}w-\frac{1}{10}x^{3}yw^{2}+\frac{1}{50}x^{2}y^{4}-\frac{1}{10}x^{2}y^{3}w+\frac{1}{10}x^{2}y^{2}w^{2}$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{5}xy$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^3}{5^5}\cdot\frac{27644130380256404651726284xyz^{8}-360613203944799684279028768xyz^{7}w+2705069415374380101768737602xyz^{6}w^{2}-11603650543812152611292432336xyz^{5}w^{3}+31039298057890937501253220355xyz^{4}w^{4}-54527646093302248482467237096xyz^{3}w^{5}+62538158752181390718083025962xyz^{2}w^{6}-44834813671900435449169231928xyzw^{7}+15247434184017389326545814044xyw^{8}+1253288841300717086810360xz^{9}+31696465729436892210506420xz^{8}w-445898668384107938598086900xz^{7}w^{2}+1552567656397819397938466730xz^{6}w^{3}-860437781763510256543303535xz^{5}w^{4}-9267568348838988680680315415xz^{4}w^{5}+32457478281209838028863083320xz^{3}w^{6}-52711943275313142464273513550xz^{2}w^{7}+48233523814910438654980436380xzw^{8}-19349828132012245840920132060xw^{9}-4045509870056585020156978y^{2}z^{8}+13536408949633002315830866y^{2}z^{7}w+233979216986512021092355736y^{2}z^{6}w^{2}-2248244404191455101399507478y^{2}z^{5}w^{3}+8671187278140385835890773815y^{2}z^{4}w^{4}-18808215089123417579993408668y^{2}z^{3}w^{5}+24113970893246989292530340656y^{2}z^{2}w^{6}-16550657102592436731429514504y^{2}zw^{7}+4071873227820872885888982612y^{2}w^{8}+3265585240580370150074358yz^{9}-8111507651448464456105266yz^{8}w-60874562965656146277000426yz^{7}w^{2}+407969935584032075232237668yz^{6}w^{3}-126641386829746027059725390yz^{5}w^{4}-4841544581940521102263095827yz^{4}w^{5}+17838515953683693114945110994yz^{3}w^{6}-30595515227408645957740996286yz^{2}w^{7}+25549190333234788450390181728yzw^{8}-6992061194567130512912040900yw^{9}+1168355216934761512228810z^{10}+9176974486746180948667380z^{9}w-223718521483841696521110020z^{8}w^{2}+1529991786947076773033106160z^{7}w^{3}-6035229689274471016150731450z^{6}w^{4}+15600065218463762673544072610z^{5}w^{5}-27095753412495150547409657420z^{4}w^{6}+30770526494091275460930251280z^{3}w^{7}-19806059650776686859191704390z^{2}w^{8}+4661374129711420341941360600zw^{9}}{34868286268525163528xyz^{8}+32148611310541611436xyz^{7}w-2936172042822836040565xyz^{6}w^{2}+17116539288415998566612xyz^{5}w^{3}-43082586350681477534900xyz^{4}w^{4}+49157933303324478170012xyz^{3}w^{5}-10060279918969313027644xyz^{2}w^{6}-26703608198661661326008xyzw^{7}+16839290805135359033120xyw^{8}+9839591489273289290xz^{9}-385044225653802168040xz^{8}w+2586339569794058767600xz^{7}w^{2}-4535043014730734880525xz^{6}w^{3}-12930845378506954428900xz^{5}w^{4}+67464465356717359751820xz^{4}w^{5}-108494025153547768417465xz^{3}w^{6}+61822883767401538751240xz^{2}w^{7}+13932511921093687264930xzw^{8}-20107742446182372412700xw^{9}-34466649640002422978y^{2}z^{8}+560064428670203898316y^{2}z^{7}w-2785868432398656776402y^{2}z^{6}w^{2}+4282475947331973014024y^{2}z^{5}w^{3}+8082868125676957463905y^{2}z^{4}w^{4}-40596314071539189162723y^{2}z^{3}w^{5}+61529987777904424429119y^{2}z^{2}w^{6}-40846940028501163677304y^{2}zw^{7}+9424542924958854016610y^{2}w^{8}+26100835589625190446yz^{9}-406525516319367709678yz^{8}w+1384881401106328995565yz^{7}w^{2}+1579956887043392493019yz^{6}w^{3}-17089064884051324377125yz^{5}w^{4}+29307842953436672976629yz^{4}w^{5}+2799874308587205576217yz^{3}w^{6}-56822405848730949276691yz^{2}w^{7}+54957255262590214420070yzw^{8}-14828238734724584074950yw^{9}-19679182978546578580z^{9}w+607476010304085324520z^{8}w^{2}-3964891022156989155120z^{7}w^{3}+9586653279939155124400z^{6}w^{4}-2664272873100046136390z^{5}w^{5}-30309262952014832669140z^{4}w^{6}+58345254727398918006290z^{3}w^{7}-41940210972423658825400z^{2}w^{8}+9885492489816389383300zw^{9}}$

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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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(7)$ $7$ $6$ $6$ $0$ $0$ full Jacobian
10.6.0.a.1 $10$ $8$ $8$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
10.6.0.a.1 $10$ $8$ $8$ $0$ $0$ full Jacobian
$X_0(14)$ $14$ $2$ $2$ $1$ $0$ $1$
70.16.0.a.1 $70$ $3$ $3$ $0$ $0$ 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
70.144.4.a.1 $70$ $3$ $3$ $4$ $0$ $2$
70.144.4.a.2 $70$ $3$ $3$ $4$ $0$ $2$
70.144.4.b.1 $70$ $3$ $3$ $4$ $0$ $1^{2}$
70.240.18.h.1 $70$ $5$ $5$ $18$ $3$ $1^{10}\cdot2^{3}$
70.288.19.i.1 $70$ $6$ $6$ $19$ $2$ $1^{11}\cdot2^{3}$
70.336.17.h.1 $70$ $7$ $7$ $17$ $4$ $1^{9}\cdot2^{3}$
70.480.35.m.1 $70$ $10$ $10$ $35$ $5$ $1^{21}\cdot2^{6}$
140.96.5.i.1 $140$ $2$ $2$ $5$ $?$ not computed
140.96.5.k.1 $140$ $2$ $2$ $5$ $?$ not computed
140.96.5.y.1 $140$ $2$ $2$ $5$ $?$ not computed
140.96.5.z.1 $140$ $2$ $2$ $5$ $?$ not computed
140.96.5.bg.1 $140$ $2$ $2$ $5$ $?$ not computed
140.96.5.bh.1 $140$ $2$ $2$ $5$ $?$ not computed
140.96.5.bo.1 $140$ $2$ $2$ $5$ $?$ not computed
140.96.5.bq.1 $140$ $2$ $2$ $5$ $?$ not computed
210.144.10.e.1 $210$ $3$ $3$ $10$ $?$ not computed
210.192.11.e.1 $210$ $4$ $4$ $11$ $?$ not computed
280.96.5.z.1 $280$ $2$ $2$ $5$ $?$ not computed
280.96.5.bf.1 $280$ $2$ $2$ $5$ $?$ not computed
280.96.5.dc.1 $280$ $2$ $2$ $5$ $?$ not computed
280.96.5.df.1 $280$ $2$ $2$ $5$ $?$ not computed
280.96.5.ea.1 $280$ $2$ $2$ $5$ $?$ not computed
280.96.5.ed.1 $280$ $2$ $2$ $5$ $?$ not computed
280.96.5.ey.1 $280$ $2$ $2$ $5$ $?$ not computed
280.96.5.fe.1 $280$ $2$ $2$ $5$ $?$ not computed