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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 4704u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4704.m3 | 4704u1 | \([0, -1, 0, -702, 6840]\) | \(5088448/441\) | \(3320525376\) | \([2, 2]\) | \(3072\) | \(0.56775\) | \(\Gamma_0(N)\)-optimal |
4704.m2 | 4704u2 | \([0, -1, 0, -2417, -37407]\) | \(3241792/567\) | \(273231802368\) | \([2]\) | \(6144\) | \(0.91432\) | |
4704.m1 | 4704u3 | \([0, -1, 0, -10992, 447252]\) | \(2438569736/21\) | \(1264962048\) | \([4]\) | \(6144\) | \(0.91432\) | |
4704.m4 | 4704u4 | \([0, -1, 0, 768, 30360]\) | \(830584/7203\) | \(-433881982464\) | \([2]\) | \(6144\) | \(0.91432\) |
Rank
sage: E.rank()
The elliptic curves in class 4704u have rank \(1\).
Complex multiplication
The elliptic curves in class 4704u do not have complex multiplication.Modular form 4704.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.