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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 59290cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
59290.by3 | 59290cd1 | \([1, 1, 0, -332147, 1433680109]\) | \(-19443408769/4249907200\) | \(-885775773781377740800\) | \([2]\) | \(3317760\) | \(2.6987\) | \(\Gamma_0(N)\)-optimal |
59290.by2 | 59290cd2 | \([1, 1, 0, -21202227, 37234215341]\) | \(5057359576472449/51765560000\) | \(10789101221839934840000\) | \([2]\) | \(6635520\) | \(3.0453\) | |
59290.by4 | 59290cd3 | \([1, 1, 0, 2988093, -38622359299]\) | \(14156681599871/3100231750000\) | \(-646157680162485625750000\) | \([2]\) | \(9953280\) | \(3.2480\) | |
59290.by1 | 59290cd4 | \([1, 1, 0, -154841887, -720668834871]\) | \(1969902499564819009/63690429687500\) | \(13274510944359854492187500\) | \([2]\) | \(19906560\) | \(3.5946\) |
Rank
sage: E.rank()
The elliptic curves in class 59290cd have rank \(0\).
Complex multiplication
The elliptic curves in class 59290cd do not have complex multiplication.Modular form 59290.2.a.cd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.