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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 222.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
222.b1 | 222c3 | \([1, 1, 0, -804, -9108]\) | \(57588477431113/78653268\) | \(78653268\) | \([2]\) | \(144\) | \(0.41989\) | |
222.b2 | 222c4 | \([1, 1, 0, -604, 5428]\) | \(24431916147913/202409388\) | \(202409388\) | \([4]\) | \(144\) | \(0.41989\) | |
222.b3 | 222c2 | \([1, 1, 0, -64, -80]\) | \(29704593673/15968016\) | \(15968016\) | \([2, 2]\) | \(72\) | \(0.073321\) | |
222.b4 | 222c1 | \([1, 1, 0, 16, 0]\) | \(410172407/255744\) | \(-255744\) | \([2]\) | \(36\) | \(-0.27325\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 222.b have rank \(0\).
Complex multiplication
The elliptic curves in class 222.b do not have complex multiplication.Modular form 222.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.