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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 7056.bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
7056.bb1 | 7056bf4 | \([0, 0, 0, -6615, -203742]\) | \(54000\) | \(592815428352\) | \([2]\) | \(6912\) | \(1.0534\) | \(-12\) | |
7056.bb2 | 7056bf2 | \([0, 0, 0, -735, 7546]\) | \(54000\) | \(813189888\) | \([2]\) | \(2304\) | \(0.50411\) | \(-12\) | |
7056.bb3 | 7056bf3 | \([0, 0, 0, 0, -9261]\) | \(0\) | \(-37050964272\) | \([2]\) | \(3456\) | \(0.70685\) | \(-3\) | |
7056.bb4 | 7056bf1 | \([0, 0, 0, 0, 343]\) | \(0\) | \(-50824368\) | \([2]\) | \(1152\) | \(0.15754\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 7056.bb have rank \(1\).
Complex multiplication
Each elliptic curve in class 7056.bb has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 7056.2.a.bb
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.