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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 912.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
912.e1 | 912b3 | \([0, -1, 0, -1272, -16560]\) | \(111223479026/3518667\) | \(7206230016\) | \([2]\) | \(384\) | \(0.66620\) | |
912.e2 | 912b2 | \([0, -1, 0, -192, 720]\) | \(768400132/263169\) | \(269485056\) | \([2, 2]\) | \(192\) | \(0.31963\) | |
912.e3 | 912b1 | \([0, -1, 0, -172, 928]\) | \(2211014608/513\) | \(131328\) | \([2]\) | \(96\) | \(-0.026945\) | \(\Gamma_0(N)\)-optimal |
912.e4 | 912b4 | \([0, -1, 0, 568, 4368]\) | \(9878111854/10097379\) | \(-20679432192\) | \([2]\) | \(384\) | \(0.66620\) |
Rank
sage: E.rank()
The elliptic curves in class 912.e have rank \(0\).
Complex multiplication
The elliptic curves in class 912.e do not have complex multiplication.Modular form 912.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.