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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 274890.bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 274890.bj1 | 274890bj8 | \([1, 1, 0, -8570252317, -247838920574531]\) | \(591720065532918583239955136329/116891407012939453125000000\) | \(13752157143665313720703125000000\) | \([2]\) | \(764411904\) | \(4.6914\) | |
| 274890.bj2 | 274890bj5 | \([1, 1, 0, -8113360882, -281290146306224]\) | \(502039459750388822744052370969/6444603154532812500\) | \(758201116527630857812500\) | \([2]\) | \(254803968\) | \(4.1420\) | |
| 274890.bj3 | 274890bj6 | \([1, 1, 0, -2638492637, 48686186772861]\) | \(17266453047612484705388895049/1288004819409000000000000\) | \(151532478998649441000000000000\) | \([2, 2]\) | \(382205952\) | \(4.3448\) | |
| 274890.bj4 | 274890bj3 | \([1, 1, 0, -2589320157, 50712515832189]\) | \(16318969429297971769640983369/102045248126976000000\) | \(12005521396890599424000000\) | \([2]\) | \(191102976\) | \(3.9982\) | |
| 274890.bj5 | 274890bj2 | \([1, 1, 0, -507521102, -4387379771676]\) | \(122884692280581205924284889/439106354595306090000\) | \(51660423511783166182410000\) | \([2, 2]\) | \(127401984\) | \(3.7955\) | |
| 274890.bj6 | 274890bj4 | \([1, 1, 0, -280626602, -8329853224776]\) | \(-20774088968758822168212889/242753662862303369030100\) | \(-28559725682087129063022234900\) | \([4]\) | \(254803968\) | \(4.1420\) | |
| 274890.bj7 | 274890bj1 | \([1, 1, 0, -46337022, 971223156]\) | \(93523304529581769096409/54118679989886265600\) | \(6367008582130129261574400\) | \([2]\) | \(63700992\) | \(3.4489\) | \(\Gamma_0(N)\)-optimal |
| 274890.bj8 | 274890bj7 | \([1, 1, 0, 2506507363, 215527217772861]\) | \(14802750729576629005731104951/179133615680899546821000000\) | \(-21074890751242150783943829000000\) | \([4]\) | \(764411904\) | \(4.6914\) |
Rank
sage: E.rank()
The elliptic curves in class 274890.bj have rank \(1\).
Complex multiplication
The elliptic curves in class 274890.bj do not have complex multiplication.Modular form 274890.2.a.bj
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.
