Show commands:
SageMath
E = EllipticCurve("gz1")
E.isogeny_class()
Elliptic curves in class 259350gz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259350.gz1 | 259350gz1 | \([1, 0, 0, -1204188, 508510992]\) | \(12359092816971484921/116188800000\) | \(1815450000000000\) | \([2]\) | \(6144000\) | \(2.0908\) | \(\Gamma_0(N)\)-optimal |
259350.gz2 | 259350gz2 | \([1, 0, 0, -1176188, 533290992]\) | \(-11516856136356002041/1201114687500000\) | \(-18767416992187500000\) | \([2]\) | \(12288000\) | \(2.4374\) |
Rank
sage: E.rank()
The elliptic curves in class 259350gz have rank \(0\).
Complex multiplication
The elliptic curves in class 259350gz do not have complex multiplication.Modular form 259350.2.a.gz
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.