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SageMath
E = EllipticCurve("db1")
E.isogeny_class()
Elliptic curves in class 33810.db
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33810.db1 | 33810dg4 | \([1, 0, 0, -220060, 39714632]\) | \(10017490085065009/235066440\) | \(27655331599560\) | \([2]\) | \(294912\) | \(1.6912\) | |
33810.db2 | 33810dg3 | \([1, 0, 0, -59340, -4991400]\) | \(196416765680689/22365315000\) | \(2631256944435000\) | \([2]\) | \(294912\) | \(1.6912\) | |
33810.db3 | 33810dg2 | \([1, 0, 0, -14260, 571472]\) | \(2725812332209/373262400\) | \(43913948097600\) | \([2, 2]\) | \(147456\) | \(1.3447\) | |
33810.db4 | 33810dg1 | \([1, 0, 0, 1420, 47760]\) | \(2691419471/9891840\) | \(-1163765084160\) | \([2]\) | \(73728\) | \(0.99810\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33810.db have rank \(1\).
Complex multiplication
The elliptic curves in class 33810.db do not have complex multiplication.Modular form 33810.2.a.db
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.