Properties

Label 3136.m
Number of curves $4$
Conductor $3136$
CM \(\Q(\sqrt{-1}) \)
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 3136.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
3136.m1 3136t3 \([0, 0, 0, -2156, -38416]\) \(287496\) \(3855122432\) \([2]\) \(1536\) \(0.70214\)   \(-16\)
3136.m2 3136t4 \([0, 0, 0, -2156, 38416]\) \(287496\) \(3855122432\) \([2]\) \(1536\) \(0.70214\)   \(-16\)
3136.m3 3136t2 \([0, 0, 0, -196, 0]\) \(1728\) \(481890304\) \([2, 2]\) \(768\) \(0.35557\)   \(-4\)
3136.m4 3136t1 \([0, 0, 0, 49, 0]\) \(1728\) \(-7529536\) \([2]\) \(384\) \(0.0089957\) \(\Gamma_0(N)\)-optimal \(-4\)

Rank

sage: E.rank()
 

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

Complex multiplication

Each elliptic curve in class 3136.m has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 3136.2.a.m

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