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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 277970bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277970.bb3 | 277970bb1 | \([1, 1, 0, -20223, -21550267]\) | \(-19443408769/4249907200\) | \(-199940628392243200\) | \([2]\) | \(3981312\) | \(1.9990\) | \(\Gamma_0(N)\)-optimal |
| 277970.bb2 | 277970bb2 | \([1, 1, 0, -1290943, -560081403]\) | \(5057359576472449/51765560000\) | \(2435356375658360000\) | \([2]\) | \(7962624\) | \(2.3456\) | |
| 277970.bb4 | 277970bb3 | \([1, 1, 0, 181937, 580360917]\) | \(14156681599871/3100231750000\) | \(-145853133982921750000\) | \([2]\) | \(11943936\) | \(2.5483\) | |
| 277970.bb1 | 277970bb4 | \([1, 1, 0, -9427883, 10822507073]\) | \(1969902499564819009/63690429687500\) | \(2996372375916992187500\) | \([2]\) | \(23887872\) | \(2.8949\) |
Rank
sage: E.rank()
The elliptic curves in class 277970bb have rank \(1\).
Complex multiplication
The elliptic curves in class 277970bb do not have complex multiplication.Modular form 277970.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
