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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 289800.cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
289800.cf1 | 289800cf5 | \([0, 0, 0, -319986075, 2203153217750]\) | \(155324313723954725282/13018359375\) | \(303692287500000000000\) | \([2]\) | \(44040192\) | \(3.3727\) | |
289800.cf2 | 289800cf4 | \([0, 0, 0, -27540075, -55569402250]\) | \(198048499826486404/242568272835\) | \(2829316334347440000000\) | \([2]\) | \(22020096\) | \(3.0261\) | |
289800.cf3 | 289800cf3 | \([0, 0, 0, -20043075, 34265384750]\) | \(76343005935514084/694180580625\) | \(8096922292410000000000\) | \([2, 2]\) | \(22020096\) | \(3.0261\) | |
289800.cf4 | 289800cf6 | \([0, 0, 0, -5868075, 81794159750]\) | \(-957928673903042/123339801817575\) | \(-2877270896800389600000000\) | \([2]\) | \(44040192\) | \(3.3727\) | |
289800.cf5 | 289800cf2 | \([0, 0, 0, -2182575, -366124750]\) | \(394315384276816/208332909225\) | \(607498763300100000000\) | \([2, 2]\) | \(11010048\) | \(2.6795\) | |
289800.cf6 | 289800cf1 | \([0, 0, 0, 518550, -44690875]\) | \(84611246065664/53699121315\) | \(-9786664859658750000\) | \([2]\) | \(5505024\) | \(2.3330\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 289800.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 289800.cf do not have complex multiplication.Modular form 289800.2.a.cf
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.