Show commands:
SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 67600cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67600.u2 | 67600cf1 | \([0, 1, 0, -2198408, -1247916812]\) | \(3803721481/26000\) | \(8031810176000000000\) | \([2]\) | \(2322432\) | \(2.4615\) | \(\Gamma_0(N)\)-optimal |
67600.u3 | 67600cf2 | \([0, 1, 0, -846408, -2764860812]\) | \(-217081801/10562500\) | \(-3262922884000000000000\) | \([2]\) | \(4644864\) | \(2.8081\) | |
67600.u1 | 67600cf3 | \([0, 1, 0, -14028408, 19407263188]\) | \(988345570681/44994560\) | \(13899529418178560000000\) | \([2]\) | \(6967296\) | \(3.0108\) | |
67600.u4 | 67600cf4 | \([0, 1, 0, 7603592, 73876639188]\) | \(157376536199/7722894400\) | \(-2385723916542054400000000\) | \([2]\) | \(13934592\) | \(3.3574\) |
Rank
sage: E.rank()
The elliptic curves in class 67600cf have rank \(0\).
Complex multiplication
The elliptic curves in class 67600cf do not have complex multiplication.Modular form 67600.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.