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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 58870a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58870.d4 | 58870a1 | \([1, -1, 0, 1945, -53075]\) | \(1367631/2800\) | \(-1665505298800\) | \([2]\) | \(96768\) | \(1.0295\) | \(\Gamma_0(N)\)-optimal |
58870.d3 | 58870a2 | \([1, -1, 0, -14875, -561039]\) | \(611960049/122500\) | \(72865856822500\) | \([2, 2]\) | \(193536\) | \(1.3761\) | |
58870.d2 | 58870a3 | \([1, -1, 0, -73745, 7221575]\) | \(74565301329/5468750\) | \(3252940036718750\) | \([2]\) | \(387072\) | \(1.7226\) | |
58870.d1 | 58870a4 | \([1, -1, 0, -225125, -41055189]\) | \(2121328796049/120050\) | \(71408539686050\) | \([2]\) | \(387072\) | \(1.7226\) |
Rank
sage: E.rank()
The elliptic curves in class 58870a have rank \(1\).
Complex multiplication
The elliptic curves in class 58870a do not have complex multiplication.Modular form 58870.2.a.a
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.