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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 89232.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
89232.o1 | 89232z4 | \([0, -1, 0, -176234608, 900467727808]\) | \(30618029936661765625/3678951124992\) | \(72735106665150507122688\) | \([2]\) | \(13934592\) | \(3.4114\) | |
89232.o2 | 89232z3 | \([0, -1, 0, -10100848, 16503217600]\) | \(-5764706497797625/2612665516032\) | \(-51653990100061806133248\) | \([2]\) | \(6967296\) | \(3.0649\) | |
89232.o3 | 89232z2 | \([0, -1, 0, -4868608, -2354164736]\) | \(645532578015625/252306960048\) | \(4988262422619450703872\) | \([2]\) | \(4644864\) | \(2.8621\) | |
89232.o4 | 89232z1 | \([0, -1, 0, 972032, -265551872]\) | \(5137417856375/4510142208\) | \(-89168261123499098112\) | \([2]\) | \(2322432\) | \(2.5156\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 89232.o have rank \(0\).
Complex multiplication
The elliptic curves in class 89232.o do not have complex multiplication.Modular form 89232.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.