Properties

Label 3136.z
Number of curves $6$
Conductor $3136$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3136.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3136.z1 3136y6 \([0, -1, 0, -8562913, -9641661567]\) \(2251439055699625/25088\) \(773738492592128\) \([2]\) \(55296\) \(2.4258\)  
3136.z2 3136y5 \([0, -1, 0, -534753, -150770815]\) \(-548347731625/1835008\) \(-56593444029595648\) \([2]\) \(27648\) \(2.0792\)  
3136.z3 3136y4 \([0, -1, 0, -111393, -11692351]\) \(4956477625/941192\) \(29027283136151552\) \([2]\) \(18432\) \(1.8765\)  
3136.z4 3136y2 \([0, -1, 0, -32993, 2316161]\) \(128787625/98\) \(3022415986688\) \([2]\) \(6144\) \(1.3272\)  
3136.z5 3136y1 \([0, -1, 0, -1633, 51969]\) \(-15625/28\) \(-863547424768\) \([2]\) \(3072\) \(0.98059\) \(\Gamma_0(N)\)-optimal
3136.z6 3136y3 \([0, -1, 0, 14047, -1080127]\) \(9938375/21952\) \(-677021181018112\) \([2]\) \(9216\) \(1.5299\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3136.z have rank \(1\).

Complex multiplication

The elliptic curves in class 3136.z do not have complex multiplication.

Modular form 3136.2.a.z

sage: E.q_eigenform(10)
 
\(q + 2q^{3} + q^{9} - 4q^{13} - 6q^{17} - 2q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.