Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 28322q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28322.z1 | 28322q1 | \([1, -1, 1, -10757935, -12827005257]\) | \(9869198625/614656\) | \(8575523825119058624768\) | \([2]\) | \(1671168\) | \(2.9600\) | \(\Gamma_0(N)\)-optimal |
28322.z2 | 28322q2 | \([1, -1, 1, 8501025, -53740739881]\) | \(4869777375/92236816\) | \(-1286864544006928734879248\) | \([2]\) | \(3342336\) | \(3.3066\) |
Rank
sage: E.rank()
The elliptic curves in class 28322q have rank \(1\).
Complex multiplication
The elliptic curves in class 28322q do not have complex multiplication.Modular form 28322.2.a.q
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.