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SageMath
E = EllipticCurve("pc1")
E.isogeny_class()
Elliptic curves in class 310464.pc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
310464.pc1 | 310464pc4 | \([0, 0, 0, -6393324, -6219544912]\) | \(1285429208617/614922\) | \(13825336844631932928\) | \([2]\) | \(9437184\) | \(2.6276\) | |
310464.pc2 | 310464pc3 | \([0, 0, 0, -3570924, 2554054832]\) | \(223980311017/4278582\) | \(96195675821289504768\) | \([2]\) | \(9437184\) | \(2.6276\) | |
310464.pc3 | 310464pc2 | \([0, 0, 0, -466284, -62535760]\) | \(498677257/213444\) | \(4798877251855712256\) | \([2, 2]\) | \(4718592\) | \(2.2811\) | |
310464.pc4 | 310464pc1 | \([0, 0, 0, 98196, -7216720]\) | \(4657463/3696\) | \(-83097441590575104\) | \([2]\) | \(2359296\) | \(1.9345\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 310464.pc have rank \(0\).
Complex multiplication
The elliptic curves in class 310464.pc do not have complex multiplication.Modular form 310464.2.a.pc
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.