Show commands:
SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 33600.cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.cr1 | 33600bd4 | \([0, -1, 0, -173633, 26275137]\) | \(282678688658/18600435\) | \(38093690880000000\) | \([2]\) | \(294912\) | \(1.9312\) | |
33600.cr2 | 33600bd2 | \([0, -1, 0, -33633, -1864863]\) | \(4108974916/893025\) | \(914457600000000\) | \([2, 2]\) | \(147456\) | \(1.5846\) | |
33600.cr3 | 33600bd1 | \([0, -1, 0, -31633, -2154863]\) | \(13674725584/945\) | \(241920000000\) | \([2]\) | \(73728\) | \(1.2380\) | \(\Gamma_0(N)\)-optimal |
33600.cr4 | 33600bd3 | \([0, -1, 0, 74367, -11476863]\) | \(22208984782/40516875\) | \(-82978560000000000\) | \([2]\) | \(294912\) | \(1.9312\) |
Rank
sage: E.rank()
The elliptic curves in class 33600.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 33600.cr do not have complex multiplication.Modular form 33600.2.a.cr
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.