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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 101430ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 101430.cd4 | 101430ca1 | \([1, -1, 0, 127175571, 1003257593253]\) | \(2652277923951208297919/6605028468326400000\) | \(-566487670822926689894400000\) | \([2]\) | \(58982400\) | \(3.8173\) | \(\Gamma_0(N)\)-optimal |
| 101430.cd3 | 101430ca2 | \([1, -1, 0, -1067264109, 11207833555365]\) | \(1567558142704512417614401/274462175610000000000\) | \(23539556163290508810000000000\) | \([2, 2]\) | \(117964800\) | \(4.1639\) | |
| 101430.cd2 | 101430ca3 | \([1, -1, 0, -4963798989, -124132071045627]\) | \(157706830105239346386477121/13650704956054687500000\) | \(1170768012996286010742187500000\) | \([2]\) | \(235929600\) | \(4.5104\) | |
| 101430.cd1 | 101430ca4 | \([1, -1, 0, -16281764109, 799626266455365]\) | \(5565604209893236690185614401/229307220930246900000\) | \(19666790856477288185274900000\) | \([2]\) | \(235929600\) | \(4.5104\) |
Rank
sage: E.rank()
The elliptic curves in class 101430ca have rank \(0\).
Complex multiplication
The elliptic curves in class 101430ca do not have complex multiplication.Modular form 101430.2.a.ca
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.
