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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 271440x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
271440.x4 | 271440x1 | \([0, 0, 0, -1488243, 698650162]\) | \(122083727651299441/32242728960\) | \(96276272790896640\) | \([2]\) | \(3932160\) | \(2.2432\) | \(\Gamma_0(N)\)-optimal |
271440.x3 | 271440x2 | \([0, 0, 0, -1672563, 514661938]\) | \(173294065906331761/61964605497600\) | \(185025320582145638400\) | \([2, 2]\) | \(7864320\) | \(2.5898\) | |
271440.x6 | 271440x3 | \([0, 0, 0, 5066637, 3618737458]\) | \(4817210305461175439/4682306425314960\) | \(-13981292069087665520640\) | \([2]\) | \(15728640\) | \(2.9364\) | |
271440.x2 | 271440x4 | \([0, 0, 0, -11360883, -14364659918]\) | \(54309086480107021681/1575939143610000\) | \(4705729067793162240000\) | \([2, 2]\) | \(15728640\) | \(2.9364\) | |
271440.x5 | 271440x5 | \([0, 0, 0, 2753997, -47701183502]\) | \(773618103830753999/329643718157812500\) | \(-984310868119737600000000\) | \([2]\) | \(31457280\) | \(3.2830\) | |
271440.x1 | 271440x6 | \([0, 0, 0, -180488883, -933304735118]\) | \(217764763259392950709681/191615146362900\) | \(572159761197277593600\) | \([2]\) | \(31457280\) | \(3.2830\) |
Rank
sage: E.rank()
The elliptic curves in class 271440x have rank \(0\).
Complex multiplication
The elliptic curves in class 271440x do not have complex multiplication.Modular form 271440.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.