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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 180999.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
180999.d1 | 180999d1 | \([1, -1, 1, -692087, -221319250]\) | \(10418796526321/6390657\) | \(22487084447440977\) | \([2]\) | \(3010560\) | \(2.0804\) | \(\Gamma_0(N)\)-optimal |
180999.d2 | 180999d2 | \([1, -1, 1, -562802, -306647350]\) | \(-5602762882081/8312741073\) | \(-29250405787427195553\) | \([2]\) | \(6021120\) | \(2.4270\) |
Rank
sage: E.rank()
The elliptic curves in class 180999.d have rank \(2\).
Complex multiplication
The elliptic curves in class 180999.d do not have complex multiplication.Modular form 180999.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.