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SageMath
E = EllipticCurve("fu1")
E.isogeny_class()
Elliptic curves in class 69696.fu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69696.fu1 | 69696gh4 | \([0, 0, 0, -2057484, -1135779568]\) | \(5690357426/891\) | \(150824283062796288\) | \([2]\) | \(983040\) | \(2.3072\) | |
69696.fu2 | 69696gh2 | \([0, 0, 0, -140844, -14161840]\) | \(3650692/1089\) | \(92170395205042176\) | \([2, 2]\) | \(491520\) | \(1.9607\) | |
69696.fu3 | 69696gh1 | \([0, 0, 0, -53724, 4621232]\) | \(810448/33\) | \(698260569735168\) | \([2]\) | \(245760\) | \(1.6141\) | \(\Gamma_0(N)\)-optimal |
69696.fu4 | 69696gh3 | \([0, 0, 0, 381876, -94660720]\) | \(36382894/43923\) | \(-7435078546540068864\) | \([2]\) | \(983040\) | \(2.3072\) |
Rank
sage: E.rank()
The elliptic curves in class 69696.fu have rank \(0\).
Complex multiplication
The elliptic curves in class 69696.fu do not have complex multiplication.Modular form 69696.2.a.fu
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.