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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 27720.y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27720.y1 | 27720p4 | \([0, 0, 0, -267987, -53388466]\) | \(1425631925916578/270703125\) | \(404157600000000\) | \([2]\) | \(131072\) | \(1.8039\) | |
27720.y2 | 27720p3 | \([0, 0, 0, -117507, 15013406]\) | \(120186986927618/4332064275\) | \(6467737306060800\) | \([2]\) | \(131072\) | \(1.8039\) | |
27720.y3 | 27720p2 | \([0, 0, 0, -18507, -648394]\) | \(939083699236/300155625\) | \(224064973440000\) | \([2, 2]\) | \(65536\) | \(1.4573\) | |
27720.y4 | 27720p1 | \([0, 0, 0, 3273, -69046]\) | \(20777545136/23059575\) | \(-4303470124800\) | \([2]\) | \(32768\) | \(1.1107\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27720.y have rank \(1\).
Complex multiplication
The elliptic curves in class 27720.y do not have complex multiplication.Modular form 27720.2.a.y
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.