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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 33600.dr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.dr1 | 33600ba4 | \([0, -1, 0, -2150433, 1214488737]\) | \(268498407453697/252\) | \(1032192000000\) | \([2]\) | \(393216\) | \(2.0335\) | |
33600.dr2 | 33600ba6 | \([0, -1, 0, -1462433, -673687263]\) | \(84448510979617/933897762\) | \(3825245233152000000\) | \([2]\) | \(786432\) | \(2.3801\) | |
33600.dr3 | 33600ba3 | \([0, -1, 0, -166433, 9304737]\) | \(124475734657/63011844\) | \(258096513024000000\) | \([2, 2]\) | \(393216\) | \(2.0335\) | |
33600.dr4 | 33600ba2 | \([0, -1, 0, -134433, 19000737]\) | \(65597103937/63504\) | \(260112384000000\) | \([2, 2]\) | \(196608\) | \(1.6870\) | |
33600.dr5 | 33600ba1 | \([0, -1, 0, -6433, 440737]\) | \(-7189057/16128\) | \(-66060288000000\) | \([2]\) | \(98304\) | \(1.3404\) | \(\Gamma_0(N)\)-optimal |
33600.dr6 | 33600ba5 | \([0, -1, 0, 617567, 71240737]\) | \(6359387729183/4218578658\) | \(-17279298183168000000\) | \([2]\) | \(786432\) | \(2.3801\) |
Rank
sage: E.rank()
The elliptic curves in class 33600.dr have rank \(0\).
Complex multiplication
The elliptic curves in class 33600.dr do not have complex multiplication.Modular form 33600.2.a.dr
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.