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SageMath
E = EllipticCurve("fp1")
E.isogeny_class()
Elliptic curves in class 304200.fp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
304200.fp1 | 304200fp2 | \([0, 0, 0, -8935875, -10010081250]\) | \(5606442/169\) | \(2378670782436000000000\) | \([2]\) | \(17203200\) | \(2.8782\) | |
304200.fp2 | 304200fp1 | \([0, 0, 0, -1330875, 370743750]\) | \(37044/13\) | \(91487337786000000000\) | \([2]\) | \(8601600\) | \(2.5316\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 304200.fp have rank \(1\).
Complex multiplication
The elliptic curves in class 304200.fp do not have complex multiplication.Modular form 304200.2.a.fp
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.