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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 105966.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
105966.d1 | 105966bb2 | \([1, -1, 0, -884941146, 10131464200756]\) | \(249896037845497/37933056\) | \(11633900348438178631962624\) | \([]\) | \(66147840\) | \(3.8223\) | |
105966.d2 | 105966bb1 | \([1, -1, 0, -25594731, -30307156619]\) | \(6045996937/2204496\) | \(676109164063411369794384\) | \([]\) | \(22049280\) | \(3.2730\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 105966.d have rank \(1\).
Complex multiplication
The elliptic curves in class 105966.d do not have complex multiplication.Modular form 105966.2.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 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.