Properties

Label 7056.bg
Number of curves $4$
Conductor $7056$
CM \(\Q(\sqrt{-7}) \)
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 7056.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
7056.bg1 7056bq4 \([0, 0, 0, -262395, 51731946]\) \(16581375\) \(120495224844288\) \([2]\) \(28672\) \(1.7628\)   \(-28\)
7056.bg2 7056bq3 \([0, 0, 0, -15435, 907578]\) \(-3375\) \(-120495224844288\) \([2]\) \(14336\) \(1.4163\)   \(-7\)
7056.bg3 7056bq2 \([0, 0, 0, -5355, -150822]\) \(16581375\) \(1024192512\) \([2]\) \(4096\) \(0.78989\)   \(-28\)
7056.bg4 7056bq1 \([0, 0, 0, -315, -2646]\) \(-3375\) \(-1024192512\) \([2]\) \(2048\) \(0.44332\) \(\Gamma_0(N)\)-optimal \(-7\)

Rank

sage: E.rank()
 

The elliptic curves in class 7056.bg have rank \(0\).

Complex multiplication

Each elliptic curve in class 7056.bg has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).

Modular form 7056.2.a.bg

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.