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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 13328w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13328.e4 | 13328w1 | \([0, 1, 0, -2368, 26676]\) | \(3048625/1088\) | \(524296650752\) | \([2]\) | \(13824\) | \(0.94971\) | \(\Gamma_0(N)\)-optimal |
13328.e3 | 13328w2 | \([0, 1, 0, -33728, 2372404]\) | \(8805624625/2312\) | \(1114130382848\) | \([2]\) | \(27648\) | \(1.2963\) | |
13328.e2 | 13328w3 | \([0, 1, 0, -80768, -8860748]\) | \(120920208625/19652\) | \(9470108254208\) | \([2]\) | \(41472\) | \(1.4990\) | |
13328.e1 | 13328w4 | \([0, 1, 0, -88608, -7045004]\) | \(159661140625/48275138\) | \(23263320926461952\) | \([2]\) | \(82944\) | \(1.8456\) |
Rank
sage: E.rank()
The elliptic curves in class 13328w have rank \(2\).
Complex multiplication
The elliptic curves in class 13328w do not have complex multiplication.Modular form 13328.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.