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SageMath
E = EllipticCurve("ki1")
E.isogeny_class()
Elliptic curves in class 177450ki
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177450.bl3 | 177450ki1 | \([1, 1, 0, -238800, -43776000]\) | \(19968681097/628992\) | \(47437878852000000\) | \([2]\) | \(2064384\) | \(1.9747\) | \(\Gamma_0(N)\)-optimal |
177450.bl2 | 177450ki2 | \([1, 1, 0, -576800, 107310000]\) | \(281397674377/96589584\) | \(7284679271210250000\) | \([2, 2]\) | \(4128768\) | \(2.3213\) | |
177450.bl1 | 177450ki3 | \([1, 1, 0, -8266300, 9142472500]\) | \(828279937799497/193444524\) | \(14589371397561187500\) | \([2]\) | \(8257536\) | \(2.6678\) | |
177450.bl4 | 177450ki4 | \([1, 1, 0, 1704700, 748411500]\) | \(7264187703863/7406095788\) | \(-558559528193445187500\) | \([2]\) | \(8257536\) | \(2.6678\) |
Rank
sage: E.rank()
The elliptic curves in class 177450ki have rank \(1\).
Complex multiplication
The elliptic curves in class 177450ki do not have complex multiplication.Modular form 177450.2.a.ki
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.