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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 82800bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82800.dk3 | 82800bg1 | \([0, 0, 0, -26175, -1629250]\) | \(680136784/345\) | \(1006020000000\) | \([2]\) | \(147456\) | \(1.2551\) | \(\Gamma_0(N)\)-optimal |
82800.dk2 | 82800bg2 | \([0, 0, 0, -30675, -1030750]\) | \(273671716/119025\) | \(1388307600000000\) | \([2, 2]\) | \(294912\) | \(1.6017\) | |
82800.dk4 | 82800bg3 | \([0, 0, 0, 104325, -7645750]\) | \(5382838942/4197615\) | \(-97921962720000000\) | \([4]\) | \(589824\) | \(1.9483\) | |
82800.dk1 | 82800bg4 | \([0, 0, 0, -237675, 43888250]\) | \(63649751618/1164375\) | \(27162540000000000\) | \([2]\) | \(589824\) | \(1.9483\) |
Rank
sage: E.rank()
The elliptic curves in class 82800bg have rank \(1\).
Complex multiplication
The elliptic curves in class 82800bg do not have complex multiplication.Modular form 82800.2.a.bg
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 Cremona 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 Cremona labels.