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SageMath
E = EllipticCurve("iv1")
E.isogeny_class()
Elliptic curves in class 486720.iv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.iv1 | 486720iv3 | \([0, 0, 0, -7643532, 8132520656]\) | \(428320044872/73125\) | \(8431473050357760000\) | \([2]\) | \(16515072\) | \(2.6381\) | \(\Gamma_0(N)\)-optimal* |
486720.iv2 | 486720iv4 | \([0, 0, 0, -3263052, -2191832656]\) | \(33324076232/1285245\) | \(148191570333087989760\) | \([2]\) | \(16515072\) | \(2.6381\) | |
486720.iv3 | 486720iv2 | \([0, 0, 0, -525252, 100253504]\) | \(1111934656/342225\) | \(4932411734459289600\) | \([2, 2]\) | \(8257536\) | \(2.2915\) | \(\Gamma_0(N)\)-optimal* |
486720.iv4 | 486720iv1 | \([0, 0, 0, 90753, 10563176]\) | \(367061696/426465\) | \(-96039747714231360\) | \([2]\) | \(4128768\) | \(1.9450\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720.iv have rank \(0\).
Complex multiplication
The elliptic curves in class 486720.iv do not have complex multiplication.Modular form 486720.2.a.iv
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.