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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 189618m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
189618.y3 | 189618m1 | \([1, 1, 1, -1399239, -573809475]\) | \(62768149033310713/6915442583808\) | \(33379520502507708672\) | \([2]\) | \(7372800\) | \(2.4796\) | \(\Gamma_0(N)\)-optimal |
189618.y2 | 189618m2 | \([1, 1, 1, -5306519, 4088357021]\) | \(3423676911662954233/483711578981136\) | \(2334783402830358075024\) | \([2, 2]\) | \(14745600\) | \(2.8262\) | |
189618.y1 | 189618m3 | \([1, 1, 1, -81785779, 284644874405]\) | \(12534210458299016895673/315581882565708\) | \(1523253471005102465772\) | \([2]\) | \(29491200\) | \(3.1727\) | |
189618.y4 | 189618m4 | \([1, 1, 1, 8656261, 21988640981]\) | \(14861225463775641287/51859390496937804\) | \(-250315372785133864787436\) | \([2]\) | \(29491200\) | \(3.1727\) |
Rank
sage: E.rank()
The elliptic curves in class 189618m have rank \(0\).
Complex multiplication
The elliptic curves in class 189618m do not have complex multiplication.Modular form 189618.2.a.m
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 Cremona 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 Cremona labels.