Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 316239.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
316239.k1 | 316239k1 | \([0, 1, 1, -789, 8345]\) | \(-39728447488/392931\) | \(-537922539\) | \([]\) | \(138240\) | \(0.49472\) | \(\Gamma_0(N)\)-optimal |
316239.k2 | 316239k2 | \([0, 1, 1, 2541, 45974]\) | \(1324839698432/1409317371\) | \(-1929355480899\) | \([]\) | \(414720\) | \(1.0440\) |
Rank
sage: E.rank()
The elliptic curves in class 316239.k have rank \(1\).
Complex multiplication
The elliptic curves in class 316239.k do not have complex multiplication.Modular form 316239.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.