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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 135240.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
135240.o1 | 135240by4 | \([0, -1, 0, -8416256, -9395001300]\) | \(273629163383866082/26408025\) | \(6362885597644800\) | \([2]\) | \(3932160\) | \(2.4662\) | |
135240.o2 | 135240by3 | \([0, -1, 0, -932976, 109990476]\) | \(372749784765122/194143359375\) | \(46777901234400000000\) | \([2]\) | \(3932160\) | \(2.4662\) | |
135240.o3 | 135240by2 | \([0, -1, 0, -527256, -145937700]\) | \(134555337776164/1312250625\) | \(158090213151360000\) | \([2, 2]\) | \(1966080\) | \(2.1196\) | |
135240.o4 | 135240by1 | \([0, -1, 0, -8836, -5549564]\) | \(-2533446736/440749575\) | \(-13274559167788800\) | \([4]\) | \(983040\) | \(1.7731\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 135240.o have rank \(0\).
Complex multiplication
The elliptic curves in class 135240.o do not have complex multiplication.Modular form 135240.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.