Properties

Label 7056.bf
Number of curves $2$
Conductor $7056$
CM \(\Q(\sqrt{-3}) \)
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 7056.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
7056.bf1 7056bh2 \([0, 0, 0, 0, -7260624]\) \(0\) \(-22773597495570432\) \([]\) \(36288\) \(1.8176\)   \(-3\)
7056.bf2 7056bh1 \([0, 0, 0, 0, 268912]\) \(0\) \(-31239502737408\) \([]\) \(12096\) \(1.2683\) \(\Gamma_0(N)\)-optimal \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 7056.bf have rank \(1\).

Complex multiplication

Each elliptic curve in class 7056.bf has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 7056.2.a.bf

sage: E.q_eigenform(10)
 
\(q + 7q^{13} - 7q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.