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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 139650.ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139650.ef1 | 139650ha5 | \([1, 0, 1, -435505901, 3497436384698]\) | \(4969327007303723277361/1123462695162150\) | \(2065222853486434146093750\) | \([2]\) | \(70778880\) | \(3.6579\) | |
| 139650.ef2 | 139650ha3 | \([1, 0, 1, -30337151, 41346947198]\) | \(1679731262160129361/570261564022500\) | \(1048292230401298476562500\) | \([2, 2]\) | \(35389440\) | \(3.3113\) | |
| 139650.ef3 | 139650ha2 | \([1, 0, 1, -12476651, -16485351802]\) | \(116844823575501841/3760263939600\) | \(6912363941093756250000\) | \([2, 2]\) | \(17694720\) | \(2.9647\) | |
| 139650.ef4 | 139650ha1 | \([1, 0, 1, -12378651, -16764259802]\) | \(114113060120923921/124104960\) | \(228137881860000000\) | \([2]\) | \(8847360\) | \(2.6181\) | \(\Gamma_0(N)\)-optimal |
| 139650.ef5 | 139650ha4 | \([1, 0, 1, 3815849, -56467146802]\) | \(3342636501165359/751262567039460\) | \(-1381020152337897336562500\) | \([2]\) | \(35389440\) | \(3.3113\) | |
| 139650.ef6 | 139650ha6 | \([1, 0, 1, 89063599, 286596087698]\) | \(42502666283088696719/43898058864843750\) | \(-80696292615468786621093750\) | \([2]\) | \(70778880\) | \(3.6579\) |
Rank
sage: E.rank()
The elliptic curves in class 139650.ef have rank \(0\).
Complex multiplication
The elliptic curves in class 139650.ef do not have complex multiplication.Modular form 139650.2.a.ef
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
