Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 16575.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16575.f1 | 16575a3 | \([1, 1, 0, -390150, -93961125]\) | \(420339554066191969/244298925\) | \(3817170703125\) | \([2]\) | \(98304\) | \(1.7389\) | |
16575.f2 | 16575a2 | \([1, 1, 0, -24525, -1458000]\) | \(104413920565969/2472575625\) | \(38633994140625\) | \([2, 2]\) | \(49152\) | \(1.3923\) | |
16575.f3 | 16575a1 | \([1, 1, 0, -3400, 41875]\) | \(278317173889/109245825\) | \(1706966015625\) | \([2]\) | \(24576\) | \(1.0457\) | \(\Gamma_0(N)\)-optimal |
16575.f4 | 16575a4 | \([1, 1, 0, 3100, -4524375]\) | \(210751100351/566398828125\) | \(-8849981689453125\) | \([2]\) | \(98304\) | \(1.7389\) |
Rank
sage: E.rank()
The elliptic curves in class 16575.f have rank \(1\).
Complex multiplication
The elliptic curves in class 16575.f do not have complex multiplication.Modular form 16575.2.a.f
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.