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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 39270.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39270.bg1 | 39270bh8 | \([1, 0, 1, -174903109, 722537464496]\) | \(591720065532918583239955136329/116891407012939453125000000\) | \(116891407012939453125000000\) | \([2]\) | \(15925248\) | \(3.7184\) | |
| 39270.bg2 | 39270bh5 | \([1, 0, 1, -165578794, 820064235992]\) | \(502039459750388822744052370969/6444603154532812500\) | \(6444603154532812500\) | \([6]\) | \(5308416\) | \(3.1691\) | |
| 39270.bg3 | 39270bh6 | \([1, 0, 1, -53846789, -141949927888]\) | \(17266453047612484705388895049/1288004819409000000000000\) | \(1288004819409000000000000\) | \([2, 2]\) | \(7962624\) | \(3.3718\) | |
| 39270.bg4 | 39270bh3 | \([1, 0, 1, -52843269, -147857449424]\) | \(16318969429297971769640983369/102045248126976000000\) | \(102045248126976000000\) | \([2]\) | \(3981312\) | \(3.0252\) | |
| 39270.bg5 | 39270bh2 | \([1, 0, 1, -10357574, 12789715016]\) | \(122884692280581205924284889/439106354595306090000\) | \(439106354595306090000\) | \([2, 6]\) | \(2654208\) | \(2.8225\) | |
| 39270.bg6 | 39270bh4 | \([1, 0, 1, -5727074, 24284468216]\) | \(-20774088968758822168212889/242753662862303369030100\) | \(-242753662862303369030100\) | \([6]\) | \(5308416\) | \(3.1691\) | |
| 39270.bg7 | 39270bh1 | \([1, 0, 1, -945654, -2966648]\) | \(93523304529581769096409/54118679989886265600\) | \(54118679989886265600\) | \([6]\) | \(1327104\) | \(2.4759\) | \(\Gamma_0(N)\)-optimal |
| 39270.bg8 | 39270bh7 | \([1, 0, 1, 51153211, -628351927888]\) | \(14802750729576629005731104951/179133615680899546821000000\) | \(-179133615680899546821000000\) | \([2]\) | \(15925248\) | \(3.7184\) |
Rank
sage: E.rank()
The elliptic curves in class 39270.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 39270.bg do not have complex multiplication.Modular form 39270.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
