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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 106722.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106722.r1 | 106722ds2 | \([1, -1, 0, -22941, 1379461]\) | \(-128667913/4096\) | \(-42507061825536\) | \([]\) | \(414720\) | \(1.3911\) | |
106722.r2 | 106722ds1 | \([1, -1, 0, 1314, 6628]\) | \(24167/16\) | \(-166043210256\) | \([]\) | \(138240\) | \(0.84175\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 106722.r have rank \(2\).
Complex multiplication
The elliptic curves in class 106722.r do not have complex multiplication.Modular form 106722.2.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 LMFDB 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 LMFDB labels.