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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 54978.bo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54978.bo1 | 54978br3 | \([1, 1, 1, -813133833, 8924310531639]\) | \(505384091400037554067434625/815656731648\) | \(95961198821655552\) | \([2]\) | \(12441600\) | \(3.4139\) | |
54978.bo2 | 54978br4 | \([1, 1, 1, -813125993, 8924491237367]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-2388653497189193005027872\) | \([2]\) | \(24883200\) | \(3.7605\) | |
54978.bo3 | 54978br1 | \([1, 1, 1, -10066953, 12165719607]\) | \(959024269496848362625/11151660319506432\) | \(1311981684929612218368\) | \([2]\) | \(4147200\) | \(2.8646\) | \(\Gamma_0(N)\)-optimal |
54978.bo4 | 54978br2 | \([1, 1, 1, -2038793, 31044740663]\) | \(-7966267523043306625/3534510366354604032\) | \(-415831610091252809760768\) | \([2]\) | \(8294400\) | \(3.2112\) |
Rank
sage: E.rank()
The elliptic curves in class 54978.bo have rank \(1\).
Complex multiplication
The elliptic curves in class 54978.bo do not have complex multiplication.Modular form 54978.2.a.bo
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.