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SageMath
E = EllipticCurve("np1")
E.isogeny_class()
Elliptic curves in class 411840.np
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
411840.np1 | 411840np4 | \([0, 0, 0, -3735372, -2778523504]\) | \(30161840495801041/2799263610\) | \(534947606479503360\) | \([2]\) | \(12582912\) | \(2.4399\) | |
411840.np2 | 411840np3 | \([0, 0, 0, -1373772, 589164176]\) | \(1500376464746641/83599963590\) | \(15976201835540643840\) | \([2]\) | \(12582912\) | \(2.4399\) | \(\Gamma_0(N)\)-optimal* |
411840.np3 | 411840np2 | \([0, 0, 0, -250572, -36682864]\) | \(9104453457841/2226896100\) | \(425566471952793600\) | \([2, 2]\) | \(6291456\) | \(2.0933\) | \(\Gamma_0(N)\)-optimal* |
411840.np4 | 411840np1 | \([0, 0, 0, 37428, -3620464]\) | \(30342134159/47190000\) | \(-9018149437440000\) | \([2]\) | \(3145728\) | \(1.7468\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 411840.np have rank \(1\).
Complex multiplication
The elliptic curves in class 411840.np do not have complex multiplication.Modular form 411840.2.a.np
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.