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SageMath
E = EllipticCurve("ee1")
E.isogeny_class()
Elliptic curves in class 103488ee
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
103488.fw4 | 103488ee1 | \([0, 1, 0, 131, -15709]\) | \(2048/891\) | \(-107341065216\) | \([2]\) | \(110592\) | \(0.79570\) | \(\Gamma_0(N)\)-optimal |
103488.fw3 | 103488ee2 | \([0, 1, 0, -8689, -306769]\) | \(37642192/1089\) | \(2099114164224\) | \([2, 2]\) | \(221184\) | \(1.1423\) | |
103488.fw2 | 103488ee3 | \([0, 1, 0, -20449, 688127]\) | \(122657188/43923\) | \(338657085161472\) | \([2]\) | \(442368\) | \(1.4888\) | |
103488.fw1 | 103488ee4 | \([0, 1, 0, -138049, -19788385]\) | \(37736227588/33\) | \(254438080512\) | \([2]\) | \(442368\) | \(1.4888\) |
Rank
sage: E.rank()
The elliptic curves in class 103488ee have rank \(0\).
Complex multiplication
The elliptic curves in class 103488ee do not have complex multiplication.Modular form 103488.2.a.ee
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.