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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 163254.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
163254.b1 | 163254cv4 | \([1, 1, 0, -26332786886, -1644736185372756]\) | \(418363773366867122598323253553/8175657796488\) | \(39462338633008446792\) | \([2]\) | \(173408256\) | \(4.2224\) | |
163254.b2 | 163254cv3 | \([1, 1, 0, -1672077046, -24836473343300]\) | \(107110600388332385155666993/6780210942198563598456\) | \(32726783197702506564099806904\) | \([2]\) | \(173408256\) | \(4.2224\) | |
163254.b3 | 163254cv2 | \([1, 1, 0, -1645800926, -25699459952460]\) | \(102139918202985795140993713/451521457781599296\) | \(2179407836113343516326464\) | \([2, 2]\) | \(86704128\) | \(3.8758\) | |
163254.b4 | 163254cv1 | \([1, 1, 0, -101222046, -415012602636]\) | \(-23762325430118066146993/1660616206139584512\) | \(-8015477249340401778782208\) | \([2]\) | \(43352064\) | \(3.5292\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 163254.b have rank \(1\).
Complex multiplication
The elliptic curves in class 163254.b do not have complex multiplication.Modular form 163254.2.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.