Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 12138.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12138.j1 | 12138m2 | \([1, 0, 1, -15945437, 24505979384]\) | \(18575453384550358633/352517816448\) | \(8508923118242934912\) | \([2]\) | \(774144\) | \(2.7560\) | |
12138.j2 | 12138m1 | \([1, 0, 1, -963677, 409316600]\) | \(-4100379159705193/626805817344\) | \(-15129568665742196736\) | \([2]\) | \(387072\) | \(2.4094\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12138.j have rank \(1\).
Complex multiplication
The elliptic curves in class 12138.j do not have complex multiplication.Modular form 12138.2.a.j
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.