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SageMath
E = EllipticCurve("kg1")
E.isogeny_class()
Elliptic curves in class 310464.kg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
310464.kg1 | 310464kg4 | \([0, 0, 0, -398890380, 3066182254352]\) | \(312196988566716625/25367712678\) | \(570344161944425473769472\) | \([2]\) | \(42467328\) | \(3.6035\) | |
310464.kg2 | 310464kg3 | \([0, 0, 0, -23228940, 54729886736]\) | \(-61653281712625/21875235228\) | \(-491822532910936381980672\) | \([2]\) | \(21233664\) | \(3.2570\) | |
310464.kg3 | 310464kg2 | \([0, 0, 0, -10245900, -6331269616]\) | \(5290763640625/2291573592\) | \(51521618699059627819008\) | \([2]\) | \(14155776\) | \(3.0542\) | |
310464.kg4 | 310464kg1 | \([0, 0, 0, 2172660, -732982768]\) | \(50447927375/39517632\) | \(-888477845486429011968\) | \([2]\) | \(7077888\) | \(2.7077\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 310464.kg have rank \(1\).
Complex multiplication
The elliptic curves in class 310464.kg do not have complex multiplication.Modular form 310464.2.a.kg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.