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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 174570.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
174570.d1 | 174570cr1 | \([1, 1, 0, -2252228, -1287755568]\) | \(8534813931497881/106867200000\) | \(15820180956940800000\) | \([2]\) | \(5068800\) | \(2.4935\) | \(\Gamma_0(N)\)-optimal |
174570.d2 | 174570cr2 | \([1, 1, 0, -390148, -3350567792]\) | \(-44365623586201/32731875000000\) | \(-4845492214261875000000\) | \([2]\) | \(10137600\) | \(2.8401\) |
Rank
sage: E.rank()
The elliptic curves in class 174570.d have rank \(1\).
Complex multiplication
The elliptic curves in class 174570.d do not have complex multiplication.Modular form 174570.2.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.