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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 291312.ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
291312.ek1 | 291312ek2 | \([0, 0, 0, -32079, -2210850]\) | \(21882096/7\) | \(1167872138496\) | \([2]\) | \(589824\) | \(1.2903\) | |
291312.ek2 | 291312ek1 | \([0, 0, 0, -1734, -44217]\) | \(-55296/49\) | \(-510944060592\) | \([2]\) | \(294912\) | \(0.94373\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 291312.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 291312.ek do not have complex multiplication.Modular form 291312.2.a.ek
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.