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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3420d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3420.d2 | 3420d1 | \([0, 0, 0, -72, 81]\) | \(3538944/1805\) | \(21053520\) | \([2]\) | \(768\) | \(0.097572\) | \(\Gamma_0(N)\)-optimal |
3420.d1 | 3420d2 | \([0, 0, 0, -927, 10854]\) | \(472058064/475\) | \(88646400\) | \([2]\) | \(1536\) | \(0.44415\) |
Rank
sage: E.rank()
The elliptic curves in class 3420d have rank \(1\).
Complex multiplication
The elliptic curves in class 3420d do not have complex multiplication.Modular form 3420.2.a.d
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.