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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 215475.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
215475.p1 | 215475w4 | \([1, 1, 1, -65935438, -206102914594]\) | \(420339554066191969/244298925\) | \(18424753904380078125\) | \([2]\) | \(16515072\) | \(3.0214\) | |
215475.p2 | 215475w2 | \([1, 1, 1, -4144813, -3182502094]\) | \(104413920565969/2472575625\) | \(186478910623916015625\) | \([2, 2]\) | \(8257536\) | \(2.6748\) | |
215475.p3 | 215475w1 | \([1, 1, 1, -574688, 94872656]\) | \(278317173889/109245825\) | \(8239198926912890625\) | \([2]\) | \(4128768\) | \(2.3282\) | \(\Gamma_0(N)\)-optimal |
215475.p4 | 215475w3 | \([1, 1, 1, 523812, -9942671094]\) | \(210751100351/566398828125\) | \(-42717171268487548828125\) | \([2]\) | \(16515072\) | \(3.0214\) |
Rank
sage: E.rank()
The elliptic curves in class 215475.p have rank \(1\).
Complex multiplication
The elliptic curves in class 215475.p do not have complex multiplication.Modular form 215475.2.a.p
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.