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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5415.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5415.e1 | 5415l4 | \([1, 0, 0, -2168715, 1229095692]\) | \(23977812996389881/146611125\) | \(6897449540026125\) | \([2]\) | \(103680\) | \(2.2264\) | |
5415.e2 | 5415l3 | \([1, 0, 0, -446745, -93110850]\) | \(209595169258201/41748046875\) | \(1964073645263671875\) | \([2]\) | \(103680\) | \(2.2264\) | |
5415.e3 | 5415l2 | \([1, 0, 0, -138090, 18437067]\) | \(6189976379881/456890625\) | \(21494821973765625\) | \([2, 2]\) | \(51840\) | \(1.8798\) | |
5415.e4 | 5415l1 | \([1, 0, 0, 8115, 1272600]\) | \(1256216039/15582375\) | \(-733086559947375\) | \([4]\) | \(25920\) | \(1.5332\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5415.e have rank \(0\).
Complex multiplication
The elliptic curves in class 5415.e do not have complex multiplication.Modular form 5415.2.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.