Show commands:
SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 139650bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139650.hm1 | 139650bj1 | \([1, 0, 0, -3199848, -2211406128]\) | \(-1231922871794037145/5186378855952\) | \(-15254307150597421200\) | \([]\) | \(4976640\) | \(2.5357\) | \(\Gamma_0(N)\)-optimal |
139650.hm2 | 139650bj2 | \([1, 0, 0, 7483377, -11658267783]\) | \(15757536948921630455/29083977048526848\) | \(-85542520394553378508800\) | \([]\) | \(14929920\) | \(3.0850\) |
Rank
sage: E.rank()
The elliptic curves in class 139650bj have rank \(0\).
Complex multiplication
The elliptic curves in class 139650bj do not have complex multiplication.Modular form 139650.2.a.bj
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 Cremona 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 Cremona labels.