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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 106470.eq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106470.eq1 | 106470ew4 | \([1, -1, 1, -407153, 100093231]\) | \(2121328796049/120050\) | \(422425188508050\) | \([2]\) | \(983040\) | \(1.8708\) | |
106470.eq2 | 106470ew3 | \([1, -1, 1, -133373, -17486153]\) | \(74565301329/5468750\) | \(19243129942968750\) | \([2]\) | \(983040\) | \(1.8708\) | |
106470.eq3 | 106470ew2 | \([1, -1, 1, -26903, 1380331]\) | \(611960049/122500\) | \(431046110722500\) | \([2, 2]\) | \(491520\) | \(1.5242\) | |
106470.eq4 | 106470ew1 | \([1, -1, 1, 3517, 127027]\) | \(1367631/2800\) | \(-9852482530800\) | \([2]\) | \(245760\) | \(1.1776\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 106470.eq have rank \(0\).
Complex multiplication
The elliptic curves in class 106470.eq do not have complex multiplication.Modular form 106470.2.a.eq
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.