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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 12342.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12342.m1 | 12342n3 | \([1, 0, 1, -2007942731, 34631635202294]\) | \(505384091400037554067434625/815656731648\) | \(1444985655175062528\) | \([2]\) | \(4147200\) | \(3.6399\) | |
12342.m2 | 12342n4 | \([1, 0, 1, -2007923371, 34632336413750]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-35968392235666975062943008\) | \([2]\) | \(8294400\) | \(3.9865\) | |
12342.m3 | 12342n1 | \([1, 0, 1, -24859211, 47222872310]\) | \(959024269496848362625/11151660319506432\) | \(19755846507285134180352\) | \([2]\) | \(1382400\) | \(3.0906\) | \(\Gamma_0(N)\)-optimal |
12342.m4 | 12342n2 | \([1, 0, 1, -5034571, 120470952182]\) | \(-7966267523043306625/3534510366354604032\) | \(-6261600719129528673533952\) | \([2]\) | \(2764800\) | \(3.4372\) |
Rank
sage: E.rank()
The elliptic curves in class 12342.m have rank \(0\).
Complex multiplication
The elliptic curves in class 12342.m do not have complex multiplication.Modular form 12342.2.a.m
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.