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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 480e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
480.a3 | 480e1 | \([0, -1, 0, -226, 1360]\) | \(20034997696/455625\) | \(29160000\) | \([2, 2]\) | \(192\) | \(0.21907\) | \(\Gamma_0(N)\)-optimal |
480.a2 | 480e2 | \([0, -1, 0, -496, -2204]\) | \(26410345352/10546875\) | \(5400000000\) | \([2]\) | \(384\) | \(0.56564\) | |
480.a1 | 480e3 | \([0, -1, 0, -3601, 84385]\) | \(1261112198464/675\) | \(2764800\) | \([4]\) | \(384\) | \(0.56564\) | |
480.a4 | 480e4 | \([0, -1, 0, 24, 3960]\) | \(2863288/13286025\) | \(-6802444800\) | \([2]\) | \(384\) | \(0.56564\) |
Rank
sage: E.rank()
The elliptic curves in class 480e have rank \(0\).
Complex multiplication
The elliptic curves in class 480e do not have complex multiplication.Modular form 480.2.a.e
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.