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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 54600.cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54600.cr1 | 54600bb4 | \([0, 1, 0, -624408, -190119312]\) | \(841356017734178/1404585\) | \(44946720000000\) | \([2]\) | \(491520\) | \(1.8819\) | |
54600.cr2 | 54600bb3 | \([0, 1, 0, -102408, 8672688]\) | \(3711757787138/1124589375\) | \(35986860000000000\) | \([2]\) | \(491520\) | \(1.8819\) | |
54600.cr3 | 54600bb2 | \([0, 1, 0, -39408, -2919312]\) | \(423026849956/16769025\) | \(268304400000000\) | \([2, 2]\) | \(245760\) | \(1.5353\) | |
54600.cr4 | 54600bb1 | \([0, 1, 0, 1092, -165312]\) | \(35969456/2985255\) | \(-11941020000000\) | \([2]\) | \(122880\) | \(1.1887\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54600.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 54600.cr do not have complex multiplication.Modular form 54600.2.a.cr
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.