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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 71478r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
71478.bb2 | 71478r1 | \([1, -1, 0, -2370213, 1045274197]\) | \(329474953/85184\) | \(380733917368195359936\) | \([]\) | \(2954880\) | \(2.6582\) | \(\Gamma_0(N)\)-optimal |
71478.bb1 | 71478r2 | \([1, -1, 0, -66879108, -210427785392]\) | \(7401701968633/2883584\) | \(12888315087108497473536\) | \([]\) | \(8864640\) | \(3.2076\) |
Rank
sage: E.rank()
The elliptic curves in class 71478r have rank \(1\).
Complex multiplication
The elliptic curves in class 71478r do not have complex multiplication.Modular form 71478.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.