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SageMath
sage: E = EllipticCurve("f1")
sage: E.isogeny_class()
Elliptic curves in class 8085.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8085.f1 | 8085g5 | \([1, 1, 1, -149410801, 702882354698]\) | \(3135316978843283198764801/571725\) | \(67262874525\) | \([2]\) | \(368640\) | \(2.8721\) | |
| 8085.f2 | 8085g4 | \([1, 1, 1, -9338176, 10979616248]\) | \(765458482133960722801/326869475625\) | \(38455866937805625\) | \([2, 2]\) | \(184320\) | \(2.5255\) | |
| 8085.f3 | 8085g6 | \([1, 1, 1, -9291871, 11093952554]\) | \(-754127868744065783521/15825714261328125\) | \(-1861879457130992578125\) | \([2]\) | \(368640\) | \(2.8721\) | |
| 8085.f4 | 8085g3 | \([1, 1, 1, -1246806, -283100196]\) | \(1821931919215868881/761147600816295\) | \(89548254088436290455\) | \([2]\) | \(184320\) | \(2.5255\) | |
| 8085.f5 | 8085g2 | \([1, 1, 1, -586531, 169584344]\) | \(189674274234120481/3859869269025\) | \(454109759631522225\) | \([2, 2]\) | \(92160\) | \(2.1789\) | |
| 8085.f6 | 8085g1 | \([1, 1, 1, 1714, 7934618]\) | \(4733169839/231139696095\) | \(-27193354105880655\) | \([2]\) | \(46080\) | \(1.8324\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8085.f have rank \(0\).
Complex multiplication
The elliptic curves in class 8085.f do not have complex multiplication.Modular form 8085.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
