Show commands:
SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 127449bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127449.o2 | 127449bc1 | \([0, 0, 1, 84966, -45920583]\) | \(32768/459\) | \(-950215219546430691\) | \([]\) | \(1327104\) | \(2.1303\) | \(\Gamma_0(N)\)-optimal |
127449.o1 | 127449bc2 | \([0, 0, 1, -7561974, -8008296858]\) | \(-23100424192/14739\) | \(-30512466494324274411\) | \([]\) | \(3981312\) | \(2.6796\) |
Rank
sage: E.rank()
The elliptic curves in class 127449bc have rank \(1\).
Complex multiplication
The elliptic curves in class 127449bc do not have complex multiplication.Modular form 127449.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.