Properties

Label 3136.o
Number of curves $4$
Conductor $3136$
CM \(\Q(\sqrt{-7}) \)
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("o1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3136.o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
3136.o1 3136r4 \([0, 0, 0, -116620, -15327984]\) \(16581375\) \(10578455953408\) \([2]\) \(7168\) \(1.5601\)   \(-28\)
3136.o2 3136r3 \([0, 0, 0, -6860, -268912]\) \(-3375\) \(-10578455953408\) \([2]\) \(3584\) \(1.2135\)   \(-7\)
3136.o3 3136r2 \([0, 0, 0, -2380, 44688]\) \(16581375\) \(89915392\) \([2]\) \(1024\) \(0.58716\)   \(-28\)
3136.o4 3136r1 \([0, 0, 0, -140, 784]\) \(-3375\) \(-89915392\) \([2]\) \(512\) \(0.24058\) \(\Gamma_0(N)\)-optimal \(-7\)

Rank

sage: E.rank()
 

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

Complex multiplication

Each elliptic curve in class 3136.o has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).

Modular form 3136.2.a.o

sage: E.q_eigenform(10)
 
\(q - 3 q^{9} + 4 q^{11} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.