Show commands:
SageMath
E = EllipticCurve("kk1")
E.isogeny_class()
Elliptic curves in class 418950.kk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
418950.kk1 | 418950kk4 | \([1, -1, 1, -6846755, -6883915503]\) | \(26487576322129/44531250\) | \(59676133996582031250\) | \([2]\) | \(18874368\) | \(2.6899\) | |
418950.kk2 | 418950kk2 | \([1, -1, 1, -562505, -34083003]\) | \(14688124849/8122500\) | \(10884926840976562500\) | \([2, 2]\) | \(9437184\) | \(2.3433\) | |
418950.kk3 | 418950kk1 | \([1, -1, 1, -342005, 76607997]\) | \(3301293169/22800\) | \(30554180606250000\) | \([2]\) | \(4718592\) | \(1.9967\) | \(\Gamma_0(N)\)-optimal* |
418950.kk4 | 418950kk3 | \([1, -1, 1, 2193745, -271120503]\) | \(871257511151/527800050\) | \(-707302546126657031250\) | \([2]\) | \(18874368\) | \(2.6899\) |
Rank
sage: E.rank()
The elliptic curves in class 418950.kk have rank \(1\).
Complex multiplication
The elliptic curves in class 418950.kk do not have complex multiplication.Modular form 418950.2.a.kk
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.