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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 19855.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19855.a1 | 19855c3 | \([1, -1, 1, -367060897, 2381187629246]\) | \(116256292809537371612841/15216540068579856875\) | \(715875533298139785538281875\) | \([2]\) | \(6359040\) | \(3.8812\) | |
19855.a2 | 19855c2 | \([1, -1, 1, -354651522, 2570738350496]\) | \(104859453317683374662841/2223652969140625\) | \(104613712971486516015625\) | \([2, 2]\) | \(3179520\) | \(3.5346\) | |
19855.a3 | 19855c1 | \([1, -1, 1, -354649717, 2570765825484]\) | \(104857852278310619039721/47155625\) | \(2218477922230625\) | \([4]\) | \(1589760\) | \(3.1880\) | \(\Gamma_0(N)\)-optimal |
19855.a4 | 19855c4 | \([1, -1, 1, -342271027, 2758530650854]\) | \(-94256762600623910012361/15323275604248046875\) | \(-720897000607656707763671875\) | \([2]\) | \(6359040\) | \(3.8812\) |
Rank
sage: E.rank()
The elliptic curves in class 19855.a have rank \(1\).
Complex multiplication
The elliptic curves in class 19855.a do not have complex multiplication.Modular form 19855.2.a.a
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.