Show commands:
SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 127050ea
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127050.cm4 | 127050ea1 | \([1, 0, 1, -14211816, 20620418278]\) | \(1433528304665250149/162339408\) | \(35949270496986000\) | \([2]\) | \(4608000\) | \(2.6007\) | \(\Gamma_0(N)\)-optimal |
127050.cm3 | 127050ea2 | \([1, 0, 1, -14248116, 20509775878]\) | \(1444540994277943589/15251205665388\) | \(3377305144972553833500\) | \([2]\) | \(9216000\) | \(2.9472\) | |
127050.cm2 | 127050ea3 | \([1, 0, 1, -52511341, -124838515672]\) | \(72313087342699809269/11447096545640448\) | \(2534903725436417212416000\) | \([2]\) | \(23040000\) | \(3.4054\) | |
127050.cm1 | 127050ea4 | \([1, 0, 1, -805228141, -8794630618072]\) | \(260744057755293612689909/8504954620259328\) | \(1883380739002654421376000\) | \([2]\) | \(46080000\) | \(3.7519\) |
Rank
sage: E.rank()
The elliptic curves in class 127050ea have rank \(0\).
Complex multiplication
The elliptic curves in class 127050ea do not have complex multiplication.Modular form 127050.2.a.ea
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.