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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 198.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
198.e1 | 198b3 | \([1, -1, 1, -725, 7661]\) | \(57736239625/255552\) | \(186297408\) | \([6]\) | \(96\) | \(0.43923\) | |
198.e2 | 198b4 | \([1, -1, 1, -365, 15005]\) | \(-7357983625/127552392\) | \(-92985693768\) | \([6]\) | \(192\) | \(0.78580\) | |
198.e3 | 198b1 | \([1, -1, 1, -50, -115]\) | \(18609625/1188\) | \(866052\) | \([2]\) | \(32\) | \(-0.11008\) | \(\Gamma_0(N)\)-optimal |
198.e4 | 198b2 | \([1, -1, 1, 40, -547]\) | \(9938375/176418\) | \(-128608722\) | \([2]\) | \(64\) | \(0.23650\) |
Rank
sage: E.rank()
The elliptic curves in class 198.e have rank \(0\).
Complex multiplication
The elliptic curves in class 198.e do not have complex multiplication.Modular form 198.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.