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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3420e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3420.b2 | 3420e1 | \([0, 0, 0, -30432, -2043331]\) | \(267219216891904/3655125\) | \(42633378000\) | \([2]\) | \(4608\) | \(1.1815\) | \(\Gamma_0(N)\)-optimal |
3420.b1 | 3420e2 | \([0, 0, 0, -31287, -1922434]\) | \(18148802937424/1947796875\) | \(363505644000000\) | \([2]\) | \(9216\) | \(1.5281\) |
Rank
sage: E.rank()
The elliptic curves in class 3420e have rank \(1\).
Complex multiplication
The elliptic curves in class 3420e do not have complex multiplication.Modular form 3420.2.a.e
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.