↑

Properties

Label 16.96.1.e.2
Level $16$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $16$ $\SL_2$-level: $16$ Newform level: $32$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $4$ are rational) Cusp widths $2^{8}\cdot4^{4}\cdot16^{4}$ Cusp orbits $1^{4}\cdot2^{2}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16M1
Rouse and Zureick-Brown (RZB) label: X468
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.96.1.34

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}13&2\\8&5\end{bmatrix}$, $\begin{bmatrix}13&4\\8&15\end{bmatrix}$, $\begin{bmatrix}13&12\\0&11\end{bmatrix}$, $\begin{bmatrix}15&14\\0&5\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $C_2\times D_4:C_4^2$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.e.2.1, 16.192.1-16.e.2.2, 16.192.1-16.e.2.3, 16.192.1-16.e.2.4, 16.192.1-16.e.2.5, 16.192.1-16.e.2.6, 16.192.1-16.e.2.7, 16.192.1-16.e.2.8, 48.192.1-16.e.2.1, 48.192.1-16.e.2.2, 48.192.1-16.e.2.3, 48.192.1-16.e.2.4, 48.192.1-16.e.2.5, 48.192.1-16.e.2.6, 48.192.1-16.e.2.7, 48.192.1-16.e.2.8, 80.192.1-16.e.2.1, 80.192.1-16.e.2.2, 80.192.1-16.e.2.3, 80.192.1-16.e.2.4, 80.192.1-16.e.2.5, 80.192.1-16.e.2.6, 80.192.1-16.e.2.7, 80.192.1-16.e.2.8, 112.192.1-16.e.2.1, 112.192.1-16.e.2.2, 112.192.1-16.e.2.3, 112.192.1-16.e.2.4, 112.192.1-16.e.2.5, 112.192.1-16.e.2.6, 112.192.1-16.e.2.7, 112.192.1-16.e.2.8, 176.192.1-16.e.2.1, 176.192.1-16.e.2.2, 176.192.1-16.e.2.3, 176.192.1-16.e.2.4, 176.192.1-16.e.2.5, 176.192.1-16.e.2.6, 176.192.1-16.e.2.7, 176.192.1-16.e.2.8, 208.192.1-16.e.2.1, 208.192.1-16.e.2.2, 208.192.1-16.e.2.3, 208.192.1-16.e.2.4, 208.192.1-16.e.2.5, 208.192.1-16.e.2.6, 208.192.1-16.e.2.7, 208.192.1-16.e.2.8, 240.192.1-16.e.2.1, 240.192.1-16.e.2.2, 240.192.1-16.e.2.3, 240.192.1-16.e.2.4, 240.192.1-16.e.2.5, 240.192.1-16.e.2.6, 240.192.1-16.e.2.7, 240.192.1-16.e.2.8, 272.192.1-16.e.2.1, 272.192.1-16.e.2.2, 272.192.1-16.e.2.3, 272.192.1-16.e.2.4, 272.192.1-16.e.2.5, 272.192.1-16.e.2.6, 272.192.1-16.e.2.7, 272.192.1-16.e.2.8, 304.192.1-16.e.2.1, 304.192.1-16.e.2.2, 304.192.1-16.e.2.3, 304.192.1-16.e.2.4, 304.192.1-16.e.2.5, 304.192.1-16.e.2.6, 304.192.1-16.e.2.7, 304.192.1-16.e.2.8
Cyclic 16-isogeny field degree: $2$
Cyclic 16-torsion field degree: $8$
Full 16-torsion field degree: $256$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 4x $
 Copy content Toggle raw display

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(0:0:1)$, $(2:-4:1)$, $(2:4:1)$

Maps to other modular curves

$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^8}\cdot\frac{192x^{2}y^{28}z^{2}-3471360x^{2}y^{24}z^{6}-26093223936x^{2}y^{20}z^{10}-64322524938240x^{2}y^{16}z^{14}-113434430646779904x^{2}y^{12}z^{18}+36262982086463324160x^{2}y^{8}z^{22}-830103506406674006016x^{2}y^{4}z^{26}+1180591550348667125760x^{2}z^{30}-32xy^{30}z-262656xy^{26}z^{5}-69599232xy^{22}z^{9}+2061188988928xy^{18}z^{13}+4785075677822976xy^{14}z^{17}-8984681707575705600xy^{10}z^{21}+553402316713728409600xy^{6}z^{25}-3246626974565067128832xy^{2}z^{29}-y^{32}+10752y^{28}z^{4}+89669632y^{24}z^{8}+204727648256y^{20}z^{12}+246287450505216y^{16}z^{16}+988539963589001216y^{12}z^{20}-96845416145140842496y^{8}z^{24}+737869666191358820352y^{4}z^{28}-281474976710656z^{32}}{z^{5}y^{16}(12x^{2}y^{8}z+66816x^{2}y^{4}z^{5}-4177920x^{2}z^{9}+xy^{10}+256xy^{6}z^{4}+3149824xy^{2}z^{8}-416y^{8}z^{3}-518144y^{4}z^{7}+65536z^{11})}$

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
8.48.0.k.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.d.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.1.b.2 $16$ $2$ $2$ $1$ $0$ 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
16.192.5.f.2 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.m.4 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.s.2 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.t.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ei.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.ej.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.em.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.en.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.288.17.fa.1 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.ir.1 $48$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
80.192.5.ho.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.hp.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.hs.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ht.2 $80$ $2$ $2$ $5$ $?$ not computed
112.192.5.ei.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.ej.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.em.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.en.1 $112$ $2$ $2$ $5$ $?$ not computed
176.192.5.ei.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.ej.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.em.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.en.1 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.ho.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.hp.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.hs.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.ht.2 $208$ $2$ $2$ $5$ $?$ not computed
240.192.5.bha.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bhb.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bhi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bhj.2 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.ho.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.hp.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.hs.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.ht.2 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.ei.1 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.ej.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.em.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.en.1 $304$ $2$ $2$ $5$ $?$ not computed