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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 222530.bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
222530.bt1 | 222530r4 | \([1, 1, 1, -1017286, -393192011]\) | \(4823468134087681/30382271150\) | \(733354166259834350\) | \([2]\) | \(5308416\) | \(2.2654\) | |
222530.bt2 | 222530r2 | \([1, 1, 1, -78036, 7997789]\) | \(2177286259681/105875000\) | \(2555565117875000\) | \([2]\) | \(1769472\) | \(1.7161\) | |
222530.bt3 | 222530r3 | \([1, 1, 1, -26016, -13337347]\) | \(-80677568161/3131816380\) | \(-75594433967580220\) | \([2]\) | \(2654208\) | \(1.9188\) | |
222530.bt4 | 222530r1 | \([1, 1, 1, 2884, 488413]\) | \(109902239/4312000\) | \(-104081197528000\) | \([2]\) | \(884736\) | \(1.3695\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 222530.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 222530.bt do not have complex multiplication.Modular form 222530.2.a.bt
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.