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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4830.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4830.i1 | 4830j3 | \([1, 0, 1, -1844, 28292]\) | \(692895692874169/51420783750\) | \(51420783750\) | \([2]\) | \(6144\) | \(0.80100\) | |
4830.i2 | 4830j2 | \([1, 0, 1, -374, -2284]\) | \(5763259856089/1143116100\) | \(1143116100\) | \([2, 2]\) | \(3072\) | \(0.45443\) | |
4830.i3 | 4830j1 | \([1, 0, 1, -354, -2588]\) | \(4886171981209/270480\) | \(270480\) | \([2]\) | \(1536\) | \(0.10786\) | \(\Gamma_0(N)\)-optimal |
4830.i4 | 4830j4 | \([1, 0, 1, 776, -13324]\) | \(51774168853511/107398242630\) | \(-107398242630\) | \([2]\) | \(6144\) | \(0.80100\) |
Rank
sage: E.rank()
The elliptic curves in class 4830.i have rank \(1\).
Complex multiplication
The elliptic curves in class 4830.i do not have complex multiplication.Modular form 4830.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.