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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 630e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 630.d4 | 630e1 | \([1, -1, 0, 21, 53]\) | \(1367631/2800\) | \(-2041200\) | \([2]\) | \(128\) | \(-0.10485\) | \(\Gamma_0(N)\)-optimal |
| 630.d3 | 630e2 | \([1, -1, 0, -159, 665]\) | \(611960049/122500\) | \(89302500\) | \([2, 2]\) | \(256\) | \(0.24172\) | |
| 630.d2 | 630e3 | \([1, -1, 0, -789, -7777]\) | \(74565301329/5468750\) | \(3986718750\) | \([2]\) | \(512\) | \(0.58829\) | |
| 630.d1 | 630e4 | \([1, -1, 0, -2409, 46115]\) | \(2121328796049/120050\) | \(87516450\) | \([2]\) | \(512\) | \(0.58829\) |
Rank
sage: E.rank()
The elliptic curves in class 630e have rank \(1\).
Complex multiplication
The elliptic curves in class 630e do not have complex multiplication.Modular form 630.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
