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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 101640bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 101640.cl5 | 101640bg1 | \([0, 1, 0, 278865, -186579450]\) | \(84611246065664/580054565475\) | \(-16441632737079303600\) | \([4]\) | \(1966080\) | \(2.3689\) | \(\Gamma_0(N)\)-optimal |
| 101640.cl4 | 101640bg2 | \([0, 1, 0, -3690540, -2490422112]\) | \(12257375872392016/1191317675625\) | \(540285934783462560000\) | \([2, 2]\) | \(3932160\) | \(2.7154\) | |
| 101640.cl3 | 101640bg3 | \([0, 1, 0, -13295520, 15897351600]\) | \(143279368983686884/22699269140625\) | \(41178255296547600000000\) | \([2, 2]\) | \(7864320\) | \(3.0620\) | |
| 101640.cl2 | 101640bg4 | \([0, 1, 0, -57596040, -168260615712]\) | \(11647843478225136004/128410942275\) | \(232947524923024665600\) | \([2]\) | \(7864320\) | \(3.0620\) | |
| 101640.cl6 | 101640bg5 | \([0, 1, 0, 23599800, 88448308848]\) | \(400647648358480318/1163177490234375\) | \(-4220190469687500000000000\) | \([2]\) | \(15728640\) | \(3.4086\) | |
| 101640.cl1 | 101640bg6 | \([0, 1, 0, -203870520, 1120317591600]\) | \(258286045443018193442/8440380939375\) | \(30623026580152554240000\) | \([2]\) | \(15728640\) | \(3.4086\) |
Rank
sage: E.rank()
The elliptic curves in class 101640bg have rank \(0\).
Complex multiplication
The elliptic curves in class 101640bg do not have complex multiplication.Modular form 101640.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
