Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 124128.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
124128.e1 | 124128i2 | \([0, 0, 0, -1740, -27072]\) | \(5268024000/185761\) | \(20543680512\) | \([2]\) | \(66560\) | \(0.74965\) | |
124128.e2 | 124128i1 | \([0, 0, 0, -1725, -27576]\) | \(328509000000/431\) | \(744768\) | \([2]\) | \(33280\) | \(0.40307\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 124128.e have rank \(1\).
Complex multiplication
The elliptic curves in class 124128.e do not have complex multiplication.Modular form 124128.2.a.e
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.