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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 117600df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117600.er3 | 117600df1 | \([0, 1, 0, -24826258, 47474097488]\) | \(14383655824793536/45209390625\) | \(5318839597640625000000\) | \([2, 2]\) | \(8847360\) | \(3.0360\) | \(\Gamma_0(N)\)-optimal |
117600.er4 | 117600df2 | \([0, 1, 0, -14407633, 87637896863]\) | \(-43927191786304/415283203125\) | \(-3126889828125000000000000\) | \([2]\) | \(17694720\) | \(3.3826\) | |
117600.er2 | 117600df3 | \([0, 1, 0, -35545008, 2476784988]\) | \(5276930158229192/3050936350875\) | \(2871516885952743000000000\) | \([2]\) | \(17694720\) | \(3.3826\) | |
117600.er1 | 117600df4 | \([0, 1, 0, -396920008, 3043572972488]\) | \(7347751505995469192/72930375\) | \(68641485507000000000\) | \([2]\) | \(17694720\) | \(3.3826\) |
Rank
sage: E.rank()
The elliptic curves in class 117600df have rank \(1\).
Complex multiplication
The elliptic curves in class 117600df do not have complex multiplication.Modular form 117600.2.a.df
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.