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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 115920er
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
115920.ep1 | 115920er1 | \([0, 0, 0, -41386827, 64425512954]\) | \(2625564132023811051529/918925030195200000\) | \(2743895437362384076800000\) | \([2]\) | \(13824000\) | \(3.3907\) | \(\Gamma_0(N)\)-optimal |
115920.ep2 | 115920er2 | \([0, 0, 0, 123763893, 450316685306]\) | \(70213095586874240921591/69970703040000000000\) | \(-208931399746191360000000000\) | \([2]\) | \(27648000\) | \(3.7373\) |
Rank
sage: E.rank()
The elliptic curves in class 115920er have rank \(1\).
Complex multiplication
The elliptic curves in class 115920er do not have complex multiplication.Modular form 115920.2.a.er
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.