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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 20286x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20286.g4 | 20286x1 | \([1, -1, 0, -264168, -85182224]\) | \(-23771111713777/22848457968\) | \(-1959623610746902128\) | \([2]\) | \(368640\) | \(2.2061\) | \(\Gamma_0(N)\)-optimal |
20286.g3 | 20286x2 | \([1, -1, 0, -4929948, -4210664900]\) | \(154502321244119857/55101928644\) | \(4725878679414669924\) | \([2, 2]\) | \(737280\) | \(2.5527\) | |
20286.g1 | 20286x3 | \([1, -1, 0, -78872418, -269590189730]\) | \(632678989847546725777/80515134\) | \(6905470724975214\) | \([2]\) | \(1474560\) | \(2.8993\) | |
20286.g2 | 20286x4 | \([1, -1, 0, -5639958, -2918020694]\) | \(231331938231569617/90942310746882\) | \(7799769227536681984722\) | \([2]\) | \(1474560\) | \(2.8993\) |
Rank
sage: E.rank()
The elliptic curves in class 20286x have rank \(1\).
Complex multiplication
The elliptic curves in class 20286x do not have complex multiplication.Modular form 20286.2.a.x
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.