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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 123210bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123210.b4 | 123210bj1 | \([1, -1, 0, -17537569965, 840631702001701]\) | \(318929057401476905525449/21353131537921474560\) | \(39939207962736320027927154524160\) | \([2]\) | \(539320320\) | \(4.8128\) | \(\Gamma_0(N)\)-optimal |
| 123210.b2 | 123210bj2 | \([1, -1, 0, -275927667885, 55787854482905125]\) | \(1242142983306846366056931529/6179359141291622400\) | \(11557963260900951434397116006400\) | \([2, 2]\) | \(1078640640\) | \(5.1594\) | |
| 123210.b3 | 123210bj3 | \([1, -1, 0, -271259487405, 57766538403876901]\) | \(-1180159344892952613848670409/87759036144023189760000\) | \(-164145778287404146771749121071360000\) | \([2]\) | \(2157281280\) | \(5.5060\) | |
| 123210.b1 | 123210bj4 | \([1, -1, 0, -4414837415085, 3570428327858577445]\) | \(5087799435928552778197163696329/125914832087040\) | \(235512934268508434572093440\) | \([2]\) | \(2157281280\) | \(5.5060\) |
Rank
sage: E.rank()
The elliptic curves in class 123210bj have rank \(0\).
Complex multiplication
The elliptic curves in class 123210bj do not have complex multiplication.Modular form 123210.2.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 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.
