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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 297689.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
297689.b1 | 297689b4 | \([1, -1, 0, -228698, -42029359]\) | \(209267191953/55223\) | \(349084631654927\) | \([2]\) | \(1505280\) | \(1.7750\) | |
297689.b2 | 297689b2 | \([1, -1, 0, -16063, -480480]\) | \(72511713/25921\) | \(163856051593129\) | \([2, 2]\) | \(752640\) | \(1.4284\) | |
297689.b3 | 297689b1 | \([1, -1, 0, -6818, 212895]\) | \(5545233/161\) | \(1017739450889\) | \([2]\) | \(376320\) | \(1.0818\) | \(\Gamma_0(N)\)-optimal |
297689.b4 | 297689b3 | \([1, -1, 0, 48652, -3418541]\) | \(2014698447/1958887\) | \(-12382835898966463\) | \([2]\) | \(1505280\) | \(1.7750\) |
Rank
sage: E.rank()
The elliptic curves in class 297689.b have rank \(0\).
Complex multiplication
The elliptic curves in class 297689.b do not have complex multiplication.Modular form 297689.2.a.b
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.