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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 11550bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11550.bl6 | 11550bk1 | \([1, 1, 1, -7163338, 7376419031]\) | \(2601656892010848045529/56330588160\) | \(880165440000000\) | \([4]\) | \(331776\) | \(2.3945\) | \(\Gamma_0(N)\)-optimal |
| 11550.bl5 | 11550bk2 | \([1, 1, 1, -7171338, 7359107031]\) | \(2610383204210122997209/12104550027662400\) | \(189133594182225000000\) | \([2, 2]\) | \(663552\) | \(2.7411\) | |
| 11550.bl4 | 11550bk3 | \([1, 1, 1, -7643713, 6330052031]\) | \(3160944030998056790089/720291785342976000\) | \(11254559145984000000000\) | \([4]\) | \(995328\) | \(2.9438\) | |
| 11550.bl3 | 11550bk4 | \([1, 1, 1, -10944338, -1228240969]\) | \(9278380528613437145689/5328033205714065000\) | \(83250518839282265625000\) | \([2]\) | \(1327104\) | \(3.0877\) | |
| 11550.bl7 | 11550bk5 | \([1, 1, 1, -3526338, 14838647031]\) | \(-310366976336070130009/5909282337130963560\) | \(-92332536517671305625000\) | \([2]\) | \(1327104\) | \(3.0877\) | |
| 11550.bl2 | 11550bk6 | \([1, 1, 1, -40411713, -93481275969]\) | \(467116778179943012100169/28800309694464000000\) | \(450004838976000000000000\) | \([2, 2]\) | \(1990656\) | \(3.2904\) | |
| 11550.bl1 | 11550bk7 | \([1, 1, 1, -636699713, -6183966907969]\) | \(1826870018430810435423307849/7641104625000000000\) | \(119392259765625000000000\) | \([2]\) | \(3981312\) | \(3.6370\) | |
| 11550.bl8 | 11550bk8 | \([1, 1, 1, 31588287, -390265275969]\) | \(223090928422700449019831/4340371122724101696000\) | \(-67818298792564089000000000\) | \([2]\) | \(3981312\) | \(3.6370\) |
Rank
sage: E.rank()
The elliptic curves in class 11550bk have rank \(0\).
Complex multiplication
The elliptic curves in class 11550bk do not have complex multiplication.Modular form 11550.2.a.bk
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
