↑

Properties

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

Related objects

Downloads

Learn more

Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $32$
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: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse and Zureick-Brown (RZB) label: X254
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.61

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}3&2\\6&5\end{bmatrix}$, $\begin{bmatrix}3&4\\0&3\end{bmatrix}$, $\begin{bmatrix}5&6\\4&1\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^3:C_4$
Contains $-I$: yes
Quadratic refinements: 8.96.1-8.c.1.1, 8.96.1-8.c.1.2, 16.96.1-8.c.1.1, 16.96.1-8.c.1.2, 16.96.1-8.c.1.3, 16.96.1-8.c.1.4, 16.96.1-8.c.1.5, 16.96.1-8.c.1.6, 24.96.1-8.c.1.1, 24.96.1-8.c.1.2, 40.96.1-8.c.1.1, 40.96.1-8.c.1.2, 48.96.1-8.c.1.1, 48.96.1-8.c.1.2, 48.96.1-8.c.1.3, 48.96.1-8.c.1.4, 48.96.1-8.c.1.5, 48.96.1-8.c.1.6, 56.96.1-8.c.1.1, 56.96.1-8.c.1.2, 80.96.1-8.c.1.1, 80.96.1-8.c.1.2, 80.96.1-8.c.1.3, 80.96.1-8.c.1.4, 80.96.1-8.c.1.5, 80.96.1-8.c.1.6, 88.96.1-8.c.1.1, 88.96.1-8.c.1.2, 104.96.1-8.c.1.1, 104.96.1-8.c.1.2, 112.96.1-8.c.1.1, 112.96.1-8.c.1.2, 112.96.1-8.c.1.3, 112.96.1-8.c.1.4, 112.96.1-8.c.1.5, 112.96.1-8.c.1.6, 120.96.1-8.c.1.1, 120.96.1-8.c.1.2, 136.96.1-8.c.1.1, 136.96.1-8.c.1.2, 152.96.1-8.c.1.1, 152.96.1-8.c.1.2, 168.96.1-8.c.1.1, 168.96.1-8.c.1.2, 176.96.1-8.c.1.1, 176.96.1-8.c.1.2, 176.96.1-8.c.1.3, 176.96.1-8.c.1.4, 176.96.1-8.c.1.5, 176.96.1-8.c.1.6, 184.96.1-8.c.1.1, 184.96.1-8.c.1.2, 208.96.1-8.c.1.1, 208.96.1-8.c.1.2, 208.96.1-8.c.1.3, 208.96.1-8.c.1.4, 208.96.1-8.c.1.5, 208.96.1-8.c.1.6, 232.96.1-8.c.1.1, 232.96.1-8.c.1.2, 240.96.1-8.c.1.1, 240.96.1-8.c.1.2, 240.96.1-8.c.1.3, 240.96.1-8.c.1.4, 240.96.1-8.c.1.5, 240.96.1-8.c.1.6, 248.96.1-8.c.1.1, 248.96.1-8.c.1.2, 264.96.1-8.c.1.1, 264.96.1-8.c.1.2, 272.96.1-8.c.1.1, 272.96.1-8.c.1.2, 272.96.1-8.c.1.3, 272.96.1-8.c.1.4, 272.96.1-8.c.1.5, 272.96.1-8.c.1.6, 280.96.1-8.c.1.1, 280.96.1-8.c.1.2, 296.96.1-8.c.1.1, 296.96.1-8.c.1.2, 304.96.1-8.c.1.1, 304.96.1-8.c.1.2, 304.96.1-8.c.1.3, 304.96.1-8.c.1.4, 304.96.1-8.c.1.5, 304.96.1-8.c.1.6, 312.96.1-8.c.1.1, 312.96.1-8.c.1.2, 328.96.1-8.c.1.1, 328.96.1-8.c.1.2
Cyclic 8-isogeny field degree: $4$
Cyclic 8-torsion field degree: $16$
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 $
 Copy content Toggle raw display

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 $j$-heightWeierstrass model
32.a3 $-4$$1728$ $= 2^{6} \cdot 3^{3}$$7.455$$(-1:0:1)$, $(0:0:1)$, $(1:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^6\,\frac{1080x^{2}y^{14}+459990x^{2}y^{12}z^{2}+5245680x^{2}y^{10}z^{4}+11896497x^{2}y^{8}z^{6}+10895400x^{2}y^{6}z^{8}+4831515x^{2}y^{4}z^{10}+1104840x^{2}y^{2}z^{12}+110565x^{2}z^{14}+16236xy^{14}z+1398160xy^{12}z^{3}+8583813xy^{10}z^{5}+16600080xy^{8}z^{7}+13010688xy^{6}z^{9}+4424760xy^{4}z^{11}+552987xy^{2}z^{13}+27y^{16}+115840y^{14}z^{2}+2741076y^{12}z^{4}+8857800y^{10}z^{6}+8890164y^{8}z^{8}+3429280y^{6}z^{10}+458442y^{4}z^{12}+1080y^{2}z^{14}+27z^{16}}{8x^{2}y^{14}+286x^{2}y^{12}z^{2}+400x^{2}y^{10}z^{4}+2013x^{2}y^{8}z^{6}+18584x^{2}y^{6}z^{8}-26625x^{2}y^{4}z^{10}+8184x^{2}y^{2}z^{12}-4095x^{2}z^{14}-4xy^{14}z+496xy^{12}z^{3}+4641xy^{10}z^{5}-4240xy^{8}z^{7}-22528xy^{6}z^{9}+32776xy^{4}z^{11}-20481xy^{2}z^{13}-y^{16}-128y^{14}z^{2}-2364y^{12}z^{4}-5384y^{10}z^{6}-7900y^{8}z^{8}+24416y^{6}z^{10}-16382y^{4}z^{12}+8y^{2}z^{14}-z^{16}}$

Modular covers

Sorry, your browser does not support the nearby lattice.

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.a.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.f.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.bc.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.bo.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.1.a.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.u.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.be.1 $8$ $2$ $2$ $1$ $0$ 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
16.96.3.b.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
16.96.3.c.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.e.1 $16$ $2$ $2$ $3$ $1$ $1^{2}$
16.96.3.e.2 $16$ $2$ $2$ $3$ $1$ $1^{2}$
16.96.3.f.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
16.96.3.g.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.5.b.1 $16$ $2$ $2$ $5$ $0$ $1^{4}$
16.96.5.c.1 $16$ $2$ $2$ $5$ $2$ $1^{4}$
24.144.9.i.1 $24$ $3$ $3$ $9$ $4$ $1^{8}$
24.192.9.g.1 $24$ $4$ $4$ $9$ $0$ $1^{8}$
40.240.17.e.1 $40$ $5$ $5$ $17$ $8$ $1^{14}\cdot2$
40.288.17.g.1 $40$ $6$ $6$ $17$ $5$ $1^{14}\cdot2$
40.480.33.i.1 $40$ $10$ $10$ $33$ $17$ $1^{28}\cdot2^{2}$
48.96.3.b.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.c.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.d.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.d.2 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.e.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.f.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.5.b.1 $48$ $2$ $2$ $5$ $0$ $1^{4}$
48.96.5.c.1 $48$ $2$ $2$ $5$ $2$ $1^{4}$
56.384.25.g.1 $56$ $8$ $8$ $25$ $6$ $1^{20}\cdot2^{2}$
56.1008.73.i.1 $56$ $21$ $21$ $73$ $39$ $1^{16}\cdot2^{26}\cdot4$
56.1344.97.i.1 $56$ $28$ $28$ $97$ $45$ $1^{36}\cdot2^{28}\cdot4$
80.96.3.b.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.c.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.d.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.d.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.e.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.f.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.5.b.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.c.1 $80$ $2$ $2$ $5$ $?$ not computed
112.96.3.b.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.c.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.d.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.d.2 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.e.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.f.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.5.b.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.c.1 $112$ $2$ $2$ $5$ $?$ not computed
176.96.3.b.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.c.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.d.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.d.2 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.e.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.f.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.5.b.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.c.1 $176$ $2$ $2$ $5$ $?$ not computed
208.96.3.b.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.c.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.d.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.d.2 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.e.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.f.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.5.b.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.c.1 $208$ $2$ $2$ $5$ $?$ not computed
240.96.3.b.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.c.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.d.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.d.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.e.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.f.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.5.b.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.c.1 $240$ $2$ $2$ $5$ $?$ not computed
272.96.3.b.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.c.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.d.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.d.2 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.e.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.f.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.5.b.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.c.1 $272$ $2$ $2$ $5$ $?$ not computed
304.96.3.b.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.c.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.d.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.d.2 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.e.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.f.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.5.b.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.c.1 $304$ $2$ $2$ $5$ $?$ not computed