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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 55440.ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55440.ef1 | 55440bm4 | \([0, 0, 0, -44427, -3604246]\) | \(12990838708516/144375\) | \(107775360000\) | \([2]\) | \(98304\) | \(1.2704\) | |
55440.ef2 | 55440bm2 | \([0, 0, 0, -2847, -53314]\) | \(13674725584/1334025\) | \(248961081600\) | \([2, 2]\) | \(49152\) | \(0.92383\) | |
55440.ef3 | 55440bm1 | \([0, 0, 0, -642, 5339]\) | \(2508888064/396165\) | \(4620868560\) | \([2]\) | \(24576\) | \(0.57725\) | \(\Gamma_0(N)\)-optimal |
55440.ef4 | 55440bm3 | \([0, 0, 0, 3453, -256174]\) | \(6099383804/41507235\) | \(-30984984898560\) | \([2]\) | \(98304\) | \(1.2704\) |
Rank
sage: E.rank()
The elliptic curves in class 55440.ef have rank \(0\).
Complex multiplication
The elliptic curves in class 55440.ef do not have complex multiplication.Modular form 55440.2.a.ef
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.