Show commands:
SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 82810.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82810.n1 | 82810bf1 | \([1, 1, 0, -82390153472, -9102542581769216]\) | \(-644487634439863642624729/896000\) | \(-85989033621056384000\) | \([]\) | \(129392640\) | \(4.4743\) | \(\Gamma_0(N)\)-optimal |
82810.n2 | 82810bf2 | \([1, 1, 0, -82368084607, -9107662613459899]\) | \(-643969879566315506524489/719323136000000000\) | \(-69033372015522001977344000000000\) | \([]\) | \(388177920\) | \(5.0236\) |
Rank
sage: E.rank()
The elliptic curves in class 82810.n have rank \(1\).
Complex multiplication
The elliptic curves in class 82810.n do not have complex multiplication.Modular form 82810.2.a.n
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.