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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 13872.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13872.f1 | 13872w4 | \([0, -1, 0, -97541064, 360095102064]\) | \(211293405175481/6973568802\) | \(3387313012059784178049024\) | \([2]\) | \(2611200\) | \(3.4800\) | |
13872.f2 | 13872w3 | \([0, -1, 0, -96754984, 366349783408]\) | \(206226044828441/236196\) | \(114728886587741847552\) | \([2]\) | \(1305600\) | \(3.1334\) | |
13872.f3 | 13872w2 | \([0, -1, 0, -13430504, -18940096656]\) | \(551569744601/2592\) | \(1259027562005397504\) | \([2]\) | \(522240\) | \(2.6753\) | |
13872.f4 | 13872w1 | \([0, -1, 0, -853224, -285474960]\) | \(141420761/9216\) | \(4476542442685857792\) | \([2]\) | \(261120\) | \(2.3287\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13872.f have rank \(0\).
Complex multiplication
The elliptic curves in class 13872.f do not have complex multiplication.Modular form 13872.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.