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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 680.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
680.b1 | 680a3 | \([0, 0, 0, -1163, -14938]\) | \(84944038338/2088025\) | \(4276275200\) | \([2]\) | \(256\) | \(0.63186\) | |
680.b2 | 680a2 | \([0, 0, 0, -163, 462]\) | \(467720676/180625\) | \(184960000\) | \([2, 2]\) | \(128\) | \(0.28529\) | |
680.b3 | 680a1 | \([0, 0, 0, -143, 658]\) | \(1263257424/425\) | \(108800\) | \([4]\) | \(64\) | \(-0.061283\) | \(\Gamma_0(N)\)-optimal |
680.b4 | 680a4 | \([0, 0, 0, 517, 3318]\) | \(7462174302/6640625\) | \(-13600000000\) | \([2]\) | \(256\) | \(0.63186\) |
Rank
sage: E.rank()
The elliptic curves in class 680.b have rank \(1\).
Complex multiplication
The elliptic curves in class 680.b do not have complex multiplication.Modular form 680.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.