Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 6450.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6450.i1 | 6450e1 | \([1, 1, 0, -4775, 115125]\) | \(770842973809/66873600\) | \(1044900000000\) | \([2]\) | \(15360\) | \(1.0470\) | \(\Gamma_0(N)\)-optimal |
6450.i2 | 6450e2 | \([1, 1, 0, 5225, 545125]\) | \(1009328859791/8734528080\) | \(-136477001250000\) | \([2]\) | \(30720\) | \(1.3936\) |
Rank
sage: E.rank()
The elliptic curves in class 6450.i have rank \(0\).
Complex multiplication
The elliptic curves in class 6450.i do not have complex multiplication.Modular form 6450.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.