Show commands:
SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 304200.ep
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
304200.ep1 | 304200ep2 | \([0, 0, 0, -162759675, 560021341750]\) | \(4234737878642/1247410125\) | \(140458131032063364000000000\) | \([2]\) | \(61931520\) | \(3.7230\) | |
304200.ep2 | 304200ep1 | \([0, 0, 0, 27365325, 58281466750]\) | \(40254822716/49359375\) | \(-2778927885249750000000000\) | \([2]\) | \(30965760\) | \(3.3765\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 304200.ep have rank \(1\).
Complex multiplication
The elliptic curves in class 304200.ep do not have complex multiplication.Modular form 304200.2.a.ep
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.