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Properties

Label 8.48.1.bv.1
Level $8$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $32$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $8^{6}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $4$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: yes $\quad(D =$ $-7$)

Other labels

Cummins and Pauli (CP) label: 8I1
Rouse and Zureick-Brown (RZB) label: X252
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.91

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}0&1\\7&0\end{bmatrix}$, $\begin{bmatrix}0&3\\7&0\end{bmatrix}$, $\begin{bmatrix}7&0\\0&3\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^2\wr C_2$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $8$
Full 8-torsion field degree: $32$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

Weierstrass model Weierstrass model

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

This modular curve has 2 rational cusps and 1 rational CM point, but no 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 $j$-heightWeierstrass model
no$\infty$ $0.000$$(1:0:1)$, $(0:1:0)$
49.a2 $-7$$-3375$ $= -1 \cdot 3^{3} \cdot 5^{3}$$8.124$$(-1:0:1)$, $(0:0:1)$

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 \frac{8x^{2}y^{25}+207x^{2}y^{24}z-6752x^{2}y^{23}z^{2}-83776x^{2}y^{22}z^{3}+237824x^{2}y^{21}z^{4}-143430x^{2}y^{20}z^{5}+4000544x^{2}y^{19}z^{6}+44607936x^{2}y^{18}z^{7}-105081392x^{2}y^{17}z^{8}+370568922x^{2}y^{16}z^{9}-1408149888x^{2}y^{15}z^{10}-1754727680x^{2}y^{14}z^{11}+3256872224x^{2}y^{13}z^{12}-9279807158x^{2}y^{12}z^{13}+42707361504x^{2}y^{11}z^{14}+62170257472x^{2}y^{10}z^{15}-154037150312x^{2}y^{9}z^{16}-171576769471x^{2}y^{8}z^{17}+174070311968x^{2}y^{7}z^{18}+229605903296x^{2}y^{6}z^{19}-47374847680x^{2}y^{5}z^{20}-134347354825x^{2}y^{4}z^{21}-28899868688x^{2}y^{3}z^{22}+24024675232x^{2}y^{2}z^{23}+12742246952x^{2}yz^{24}+1768944337x^{2}z^{25}-xy^{26}-144xy^{25}z-2656xy^{24}z^{2}+27976xy^{23}z^{3}+271056xy^{22}z^{4}-470096xy^{21}z^{5}+4763168xy^{20}z^{6}-20259152xy^{19}z^{7}-57091682xy^{18}z^{8}+67634944xy^{17}z^{9}-974414336xy^{16}z^{10}+3150170024xy^{15}z^{11}+1163645188xy^{14}z^{12}+1663710512xy^{13}z^{13}+21346304288xy^{12}z^{14}-83046602064xy^{11}z^{15}-106830899691xy^{10}z^{16}+219199230960xy^{9}z^{17}+249415368864xy^{8}z^{18}-202259067752xy^{7}z^{19}-286902321160xy^{6}z^{20}+37706858512xy^{5}z^{21}+147004361824xy^{4}z^{22}+35284631000xy^{3}z^{23}-23149408977xy^{2}z^{24}-12738101248xyz^{25}-1769996288xz^{26}+1224y^{25}z^{2}+18260y^{24}z^{3}-86672y^{23}z^{4}-456800y^{22}z^{5}+38928y^{21}z^{6}-19407188y^{20}z^{7}+58604576y^{19}z^{8}-60759872y^{18}z^{9}+397650928y^{17}z^{10}+1475030001y^{16}z^{11}-3846266832y^{15}z^{12}+3175751200y^{14}z^{13}-18559213040y^{13}z^{14}-34520063716y^{12}z^{15}+102876155168y^{11}z^{16}+113708112576y^{10}z^{17}-144942557192y^{9}z^{18}-179166073192y^{8}z^{19}+53906822096y^{7}z^{20}+121228453856y^{6}z^{21}+22550810856y^{5}z^{22}-24915090022y^{4}z^{23}-12736987120y^{3}z^{24}-1769698208y^{2}z^{25}+48600yz^{26}+3375z^{27}}{z^{3}(20x^{2}y^{22}-5997x^{2}y^{20}z^{2}+231996x^{2}y^{18}z^{4}-3031386x^{2}y^{16}z^{6}+18197648x^{2}y^{14}z^{8}-57915333x^{2}y^{12}z^{10}+105512028x^{2}y^{10}z^{12}-114556554x^{2}y^{8}z^{14}+74973028x^{2}y^{6}z^{16}-28901319x^{2}y^{4}z^{18}+6029300x^{2}y^{2}z^{20}-524287x^{2}z^{22}-194xy^{22}z+24068xy^{20}z^{3}-630310xy^{18}z^{5}+6424952xy^{16}z^{7}-32180840xy^{14}z^{9}+88999012xy^{12}z^{11}-144966066xy^{10}z^{13}+143786152xy^{8}z^{15}-87425082xy^{6}z^{17}+31719436xy^{4}z^{19}-6291457xy^{2}z^{21}+524288xz^{23}-y^{24}+1232y^{22}z^{2}-77312y^{20}z^{4}+1342816y^{18}z^{6}-9855442y^{16}z^{8}+36572736y^{14}z^{10}-75236940y^{12}z^{12}+90047216y^{10}z^{14}-63766843y^{8}z^{16}+26214544y^{6}z^{18}-5767224y^{4}z^{20}+524300y^{2}z^{22}-z^{24})}$

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
8.24.0.bt.1 $8$ $2$ $2$ $0$ $0$ 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
$X_{\mathrm{sp}}(8)$ $8$ $2$ $2$ $3$ $0$ $1^{2}$
8.96.3.l.1 $8$ $2$ $2$ $3$ $0$ $1^{2}$
8.96.3.n.1 $8$ $2$ $2$ $3$ $0$ $1^{2}$
8.96.3.p.1 $8$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.eg.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.ei.1 $16$ $2$ $2$ $3$ $1$ $1^{2}$
16.96.3.fk.1 $16$ $2$ $2$ $3$ $1$ $1^{2}$
16.96.3.fq.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
16.96.5.cs.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.96.5.cy.1 $16$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
16.96.5.ds.1 $16$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
16.96.5.du.1 $16$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.96.3.if.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.ij.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.in.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.ir.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.144.7.cov.1 $24$ $3$ $3$ $7$ $2$ $1^{6}$
24.192.11.n.1 $24$ $4$ $4$ $11$ $1$ $1^{10}$
40.96.3.cd.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.96.3.ch.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.96.3.cl.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.96.3.cp.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.240.17.bid.1 $40$ $5$ $5$ $17$ $7$ $1^{14}\cdot2$
40.288.17.cvr.1 $40$ $6$ $6$ $17$ $2$ $1^{14}\cdot2$
40.480.33.ezt.1 $40$ $10$ $10$ $33$ $15$ $1^{28}\cdot2^{2}$
48.96.3.xc.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.xe.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.xy.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.ye.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.5.qe.1 $48$ $2$ $2$ $5$ $4$ $1^{2}\cdot2$
48.96.5.qk.1 $48$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
48.96.5.re.1 $48$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
48.96.5.rg.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.96.3.bi.1 $56$ $2$ $2$ $3$ $2$ $1^{2}$
56.96.3.bm.1 $56$ $2$ $2$ $3$ $2$ $1^{2}$
56.96.3.bq.1 $56$ $2$ $2$ $3$ $2$ $1^{2}$
56.96.3.bu.1 $56$ $2$ $2$ $3$ $2$ $1^{2}$
56.384.27.p.1 $56$ $8$ $8$ $27$ $6$ $1^{18}\cdot2^{4}$
56.1008.71.zb.1 $56$ $21$ $21$ $71$ $35$ $1^{14}\cdot2^{26}\cdot4$
56.1344.97.ewr.1 $56$ $28$ $28$ $97$ $41$ $1^{32}\cdot2^{30}\cdot4$
80.96.3.bbc.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.bbe.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.bby.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.bce.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.5.ra.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.rg.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.sa.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.sc.1 $80$ $2$ $2$ $5$ $?$ not computed
88.96.3.bh.1 $88$ $2$ $2$ $3$ $?$ not computed
88.96.3.bl.1 $88$ $2$ $2$ $3$ $?$ not computed
88.96.3.bp.1 $88$ $2$ $2$ $3$ $?$ not computed
88.96.3.bt.1 $88$ $2$ $2$ $3$ $?$ not computed
104.96.3.ce.1 $104$ $2$ $2$ $3$ $?$ not computed
104.96.3.ci.1 $104$ $2$ $2$ $3$ $?$ not computed
104.96.3.cm.1 $104$ $2$ $2$ $3$ $?$ not computed
104.96.3.cq.1 $104$ $2$ $2$ $3$ $?$ not computed
112.96.3.wc.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.we.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.wy.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.xe.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.5.pm.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.ps.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.qm.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.qo.1 $112$ $2$ $2$ $5$ $?$ not computed
120.96.3.bar.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.baz.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bbh.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bbp.1 $120$ $2$ $2$ $3$ $?$ not computed
136.96.3.cd.1 $136$ $2$ $2$ $3$ $?$ not computed
136.96.3.ch.1 $136$ $2$ $2$ $3$ $?$ not computed
136.96.3.cl.1 $136$ $2$ $2$ $3$ $?$ not computed
136.96.3.cp.1 $136$ $2$ $2$ $3$ $?$ not computed
152.96.3.bi.1 $152$ $2$ $2$ $3$ $?$ not computed
152.96.3.bm.1 $152$ $2$ $2$ $3$ $?$ not computed
152.96.3.bq.1 $152$ $2$ $2$ $3$ $?$ not computed
152.96.3.bu.1 $152$ $2$ $2$ $3$ $?$ not computed
168.96.3.yd.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.yl.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.yt.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.zb.1 $168$ $2$ $2$ $3$ $?$ not computed
176.96.3.wc.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.we.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.wy.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.xe.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.5.pm.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.ps.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.qm.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.qo.1 $176$ $2$ $2$ $5$ $?$ not computed
184.96.3.bh.1 $184$ $2$ $2$ $3$ $?$ not computed
184.96.3.bl.1 $184$ $2$ $2$ $3$ $?$ not computed
184.96.3.bp.1 $184$ $2$ $2$ $3$ $?$ not computed
184.96.3.bt.1 $184$ $2$ $2$ $3$ $?$ not computed
208.96.3.bbc.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.bbe.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.bby.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.bce.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.5.ra.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.rg.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.sa.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.sc.1 $208$ $2$ $2$ $5$ $?$ not computed
232.96.3.cd.1 $232$ $2$ $2$ $3$ $?$ not computed
232.96.3.ch.1 $232$ $2$ $2$ $3$ $?$ not computed
232.96.3.cl.1 $232$ $2$ $2$ $3$ $?$ not computed
232.96.3.cp.1 $232$ $2$ $2$ $3$ $?$ not computed
240.96.3.fpi.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fpk.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fqu.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fra.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.5.dmi.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dmo.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dny.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.doa.1 $240$ $2$ $2$ $5$ $?$ not computed
248.96.3.bi.1 $248$ $2$ $2$ $3$ $?$ not computed
248.96.3.bm.1 $248$ $2$ $2$ $3$ $?$ not computed
248.96.3.bq.1 $248$ $2$ $2$ $3$ $?$ not computed
248.96.3.bu.1 $248$ $2$ $2$ $3$ $?$ not computed
264.96.3.yd.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.yl.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.yt.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.zb.1 $264$ $2$ $2$ $3$ $?$ not computed
272.96.3.bbc.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.bbe.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.bby.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.bce.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.5.ra.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.rg.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.sa.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.sc.1 $272$ $2$ $2$ $5$ $?$ not computed
280.96.3.gj.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.gr.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.gz.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.hh.1 $280$ $2$ $2$ $3$ $?$ not computed
296.96.3.ce.1 $296$ $2$ $2$ $3$ $?$ not computed
296.96.3.ci.1 $296$ $2$ $2$ $3$ $?$ not computed
296.96.3.cm.1 $296$ $2$ $2$ $3$ $?$ not computed
296.96.3.cq.1 $296$ $2$ $2$ $3$ $?$ not computed
304.96.3.wc.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.we.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.wy.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.xe.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.5.pm.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.ps.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.qm.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.qo.1 $304$ $2$ $2$ $5$ $?$ not computed
312.96.3.bar.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.baz.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bbh.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bbp.1 $312$ $2$ $2$ $3$ $?$ not computed
328.96.3.cd.1 $328$ $2$ $2$ $3$ $?$ not computed
328.96.3.ch.1 $328$ $2$ $2$ $3$ $?$ not computed
328.96.3.cl.1 $328$ $2$ $2$ $3$ $?$ not computed
328.96.3.cp.1 $328$ $2$ $2$ $3$ $?$ not computed