Show commands:
SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 21175.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21175.m1 | 21175bc2 | \([1, 1, 1, -3533263, -2417958344]\) | \(1409825840597/86806489\) | \(300357403240876953125\) | \([2]\) | \(806400\) | \(2.6811\) | |
21175.m2 | 21175bc1 | \([1, 1, 1, 172362, -157527094]\) | \(163667323/3195731\) | \(-11057485168146484375\) | \([2]\) | \(403200\) | \(2.3345\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21175.m have rank \(1\).
Complex multiplication
The elliptic curves in class 21175.m do not have complex multiplication.Modular form 21175.2.a.m
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.