Properties

Label 3136.p
Number of curves $4$
Conductor $3136$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3136.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3136.p1 3136s4 \([0, 0, 0, -58604, 5460560]\) \(1443468546/7\) \(107943428096\) \([2]\) \(6144\) \(1.3175\)  
3136.p2 3136s3 \([0, 0, 0, -11564, -378672]\) \(11090466/2401\) \(37024595836928\) \([2]\) \(6144\) \(1.3175\)  
3136.p3 3136s2 \([0, 0, 0, -3724, 82320]\) \(740772/49\) \(377801998336\) \([2, 2]\) \(3072\) \(0.97092\)  
3136.p4 3136s1 \([0, 0, 0, 196, 5488]\) \(432/7\) \(-13492928512\) \([2]\) \(1536\) \(0.62435\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3136.2.a.p

sage: E.q_eigenform(10)
 
\(q + 2 q^{5} - 3 q^{9} - 4 q^{11} + 2 q^{13} + 6 q^{17} - 8 q^{19} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.