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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 60690.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60690.bp1 | 60690bl8 | \([1, 1, 1, -1864345, 615650345]\) | \(29689921233686449/10380965400750\) | \(250571268647215776750\) | \([2]\) | \(2654208\) | \(2.6156\) | |
| 60690.bp2 | 60690bl5 | \([1, 1, 1, -1664935, 826189157]\) | \(21145699168383889/2593080\) | \(62590647422520\) | \([2]\) | \(884736\) | \(2.0663\) | |
| 60690.bp3 | 60690bl6 | \([1, 1, 1, -780595, -258719155]\) | \(2179252305146449/66177562500\) | \(1597365481095562500\) | \([2, 2]\) | \(1327104\) | \(2.2691\) | |
| 60690.bp4 | 60690bl3 | \([1, 1, 1, -774815, -262832203]\) | \(2131200347946769/2058000\) | \(49675117002000\) | \([2]\) | \(663552\) | \(1.9225\) | |
| 60690.bp5 | 60690bl2 | \([1, 1, 1, -104335, 12804437]\) | \(5203798902289/57153600\) | \(1379548963598400\) | \([2, 2]\) | \(442368\) | \(1.7198\) | |
| 60690.bp6 | 60690bl4 | \([1, 1, 1, -23415, 32257605]\) | \(-58818484369/18600435000\) | \(-448969283242515000\) | \([2]\) | \(884736\) | \(2.0663\) | |
| 60690.bp7 | 60690bl1 | \([1, 1, 1, -11855, -179755]\) | \(7633736209/3870720\) | \(93429771079680\) | \([2]\) | \(221184\) | \(1.3732\) | \(\Gamma_0(N)\)-optimal |
| 60690.bp8 | 60690bl7 | \([1, 1, 1, 210675, -869737983]\) | \(42841933504271/13565917968750\) | \(-327448281019042968750\) | \([2]\) | \(2654208\) | \(2.6156\) |
Rank
sage: E.rank()
The elliptic curves in class 60690.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 60690.bp do not have complex multiplication.Modular form 60690.2.a.bp
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.
