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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 448.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
448.e1 | 448b3 | \([0, 0, 0, -1196, -15920]\) | \(1443468546/7\) | \(917504\) | \([2]\) | \(128\) | \(0.34454\) | |
448.e2 | 448b4 | \([0, 0, 0, -236, 1104]\) | \(11090466/2401\) | \(314703872\) | \([4]\) | \(128\) | \(0.34454\) | |
448.e3 | 448b2 | \([0, 0, 0, -76, -240]\) | \(740772/49\) | \(3211264\) | \([2, 2]\) | \(64\) | \(-0.0020328\) | |
448.e4 | 448b1 | \([0, 0, 0, 4, -16]\) | \(432/7\) | \(-114688\) | \([2]\) | \(32\) | \(-0.34861\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 448.e have rank \(1\).
Complex multiplication
The elliptic curves in class 448.e do not have complex multiplication.Modular form 448.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.