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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 72450bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.bt2 | 72450bn1 | \([1, -1, 0, 75033, -18301059]\) | \(4101378352343/15049939968\) | \(-171428222448000000\) | \([2]\) | \(983040\) | \(1.9898\) | \(\Gamma_0(N)\)-optimal |
72450.bt1 | 72450bn2 | \([1, -1, 0, -752967, -219505059]\) | \(4144806984356137/568114785504\) | \(6471182478631500000\) | \([2]\) | \(1966080\) | \(2.3364\) |
Rank
sage: E.rank()
The elliptic curves in class 72450bn have rank \(1\).
Complex multiplication
The elliptic curves in class 72450bn do not have complex multiplication.Modular form 72450.2.a.bn
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.