Show commands:
SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 414736.bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414736.bj1 | 414736bj4 | \([0, 0, 0, -51297659, -141382413718]\) | \(209267191953/55223\) | \(3939446439701959897088\) | \([2]\) | \(32440320\) | \(3.1282\) | |
414736.bj2 | 414736bj2 | \([0, 0, 0, -3603019, -1627579590]\) | \(72511713/25921\) | \(1849127920676430155776\) | \([2, 2]\) | \(16220160\) | \(2.7816\) | |
414736.bj3 | 414736bj1 | \([0, 0, 0, -1529339, 709457770]\) | \(5545233/161\) | \(11485266588052361216\) | \([2]\) | \(8110080\) | \(2.4351\) | \(\Gamma_0(N)\)-optimal* |
414736.bj4 | 414736bj3 | \([0, 0, 0, 10912741, -11443136502]\) | \(2014698447/1958887\) | \(-139741238576833078915072\) | \([2]\) | \(32440320\) | \(3.1282\) |
Rank
sage: E.rank()
The elliptic curves in class 414736.bj have rank \(0\).
Complex multiplication
The elliptic curves in class 414736.bj do not have complex multiplication.Modular form 414736.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.