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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 425.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
425.d1 | 425a3 | \([1, -1, 0, -2267, -40984]\) | \(82483294977/17\) | \(265625\) | \([2]\) | \(128\) | \(0.42808\) | |
425.d2 | 425a2 | \([1, -1, 0, -142, -609]\) | \(20346417/289\) | \(4515625\) | \([2, 2]\) | \(64\) | \(0.081509\) | |
425.d3 | 425a4 | \([1, -1, 0, -17, -1734]\) | \(-35937/83521\) | \(-1305015625\) | \([2]\) | \(128\) | \(0.42808\) | |
425.d4 | 425a1 | \([1, -1, 0, -17, 16]\) | \(35937/17\) | \(265625\) | \([2]\) | \(32\) | \(-0.26506\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 425.d have rank \(1\).
Complex multiplication
The elliptic curves in class 425.d do not have complex multiplication.Modular form 425.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.