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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 33600.dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.dz1 | 33600gi4 | \([0, 1, 0, -2150433, -1214488737]\) | \(268498407453697/252\) | \(1032192000000\) | \([2]\) | \(393216\) | \(2.0335\) | |
33600.dz2 | 33600gi6 | \([0, 1, 0, -1462433, 673687263]\) | \(84448510979617/933897762\) | \(3825245233152000000\) | \([2]\) | \(786432\) | \(2.3801\) | |
33600.dz3 | 33600gi3 | \([0, 1, 0, -166433, -9304737]\) | \(124475734657/63011844\) | \(258096513024000000\) | \([2, 2]\) | \(393216\) | \(2.0335\) | |
33600.dz4 | 33600gi2 | \([0, 1, 0, -134433, -19000737]\) | \(65597103937/63504\) | \(260112384000000\) | \([2, 2]\) | \(196608\) | \(1.6870\) | |
33600.dz5 | 33600gi1 | \([0, 1, 0, -6433, -440737]\) | \(-7189057/16128\) | \(-66060288000000\) | \([2]\) | \(98304\) | \(1.3404\) | \(\Gamma_0(N)\)-optimal |
33600.dz6 | 33600gi5 | \([0, 1, 0, 617567, -71240737]\) | \(6359387729183/4218578658\) | \(-17279298183168000000\) | \([2]\) | \(786432\) | \(2.3801\) |
Rank
sage: E.rank()
The elliptic curves in class 33600.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 33600.dz do not have complex multiplication.Modular form 33600.2.a.dz
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.