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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 206310bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 206310.bi6 | 206310bg1 | \([1, 1, 1, 7924, -105427]\) | \(371694959/249600\) | \(-36949757894400\) | \([2]\) | \(720896\) | \(1.2921\) | \(\Gamma_0(N)\)-optimal |
| 206310.bi5 | 206310bg2 | \([1, 1, 1, -34396, -917971]\) | \(30400540561/15210000\) | \(2251625871690000\) | \([2, 2]\) | \(1441792\) | \(1.6387\) | |
| 206310.bi3 | 206310bg3 | \([1, 1, 1, -298896, 62138829]\) | \(19948814692561/231344100\) | \(34247229508404900\) | \([2, 2]\) | \(2883584\) | \(1.9852\) | |
| 206310.bi2 | 206310bg4 | \([1, 1, 1, -447016, -115131187]\) | \(66730743078481/60937500\) | \(9020936985937500\) | \([2]\) | \(2883584\) | \(1.9852\) | |
| 206310.bi1 | 206310bg5 | \([1, 1, 1, -4768946, 4006510949]\) | \(81025909800741361/11088090\) | \(1641435260462010\) | \([2]\) | \(5767168\) | \(2.3318\) | |
| 206310.bi4 | 206310bg6 | \([1, 1, 1, -60846, 158691909]\) | \(-168288035761/73415764890\) | \(-10868168022106137210\) | \([2]\) | \(5767168\) | \(2.3318\) |
Rank
sage: E.rank()
The elliptic curves in class 206310bg have rank \(0\).
Complex multiplication
The elliptic curves in class 206310bg do not have complex multiplication.Modular form 206310.2.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.
