Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 27930.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27930.a1 | 27930j1 | \([1, 1, 0, -7088, 226752]\) | \(114840864304543/3119040\) | \(1069830720\) | \([2]\) | \(36864\) | \(0.83657\) | \(\Gamma_0(N)\)-optimal |
27930.a2 | 27930j2 | \([1, 1, 0, -6808, 245848]\) | \(-101762531964703/19000801800\) | \(-6517275017400\) | \([2]\) | \(73728\) | \(1.1831\) |
Rank
sage: E.rank()
The elliptic curves in class 27930.a have rank \(1\).
Complex multiplication
The elliptic curves in class 27930.a do not have complex multiplication.Modular form 27930.2.a.a
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.