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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 68970bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68970.bd4 | 68970bi1 | \([1, 0, 1, -1213, -27832]\) | \(-111284641/123120\) | \(-218114590320\) | \([2]\) | \(122880\) | \(0.87032\) | \(\Gamma_0(N)\)-optimal |
68970.bd3 | 68970bi2 | \([1, 0, 1, -22993, -1343344]\) | \(758800078561/324900\) | \(575580168900\) | \([2, 2]\) | \(245760\) | \(1.2169\) | |
68970.bd2 | 68970bi3 | \([1, 0, 1, -26623, -891772]\) | \(1177918188481/488703750\) | \(865768504053750\) | \([2]\) | \(491520\) | \(1.5635\) | |
68970.bd1 | 68970bi4 | \([1, 0, 1, -367843, -85900564]\) | \(3107086841064961/570\) | \(1009789770\) | \([2]\) | \(491520\) | \(1.5635\) |
Rank
sage: E.rank()
The elliptic curves in class 68970bi have rank \(1\).
Complex multiplication
The elliptic curves in class 68970bi do not have complex multiplication.Modular form 68970.2.a.bi
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.