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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 122018f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122018.j4 | 122018f1 | \([1, -1, 0, 232597, -14656459]\) | \(6128487/3952\) | \(-897426016167377008\) | \([2]\) | \(2056320\) | \(2.1338\) | \(\Gamma_0(N)\)-optimal |
| 122018.j3 | 122018f2 | \([1, -1, 0, -987583, -119835975]\) | \(469097433/244036\) | \(55416056498335530244\) | \([2, 2]\) | \(4112640\) | \(2.4804\) | |
| 122018.j2 | 122018f3 | \([1, -1, 0, -8918753, 10166891515]\) | \(345505073913/3388346\) | \(769430630611504862234\) | \([2]\) | \(8225280\) | \(2.8269\) | |
| 122018.j1 | 122018f4 | \([1, -1, 0, -12579293, -17152694649]\) | \(969417177273/1085318\) | \(246455619689965910822\) | \([2]\) | \(8225280\) | \(2.8269\) |
Rank
sage: E.rank()
The elliptic curves in class 122018f have rank \(0\).
Complex multiplication
The elliptic curves in class 122018f do not have complex multiplication.Modular form 122018.2.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 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.
