Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 174a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
174.b1 | 174a1 | \([1, 0, 1, -7705, 1226492]\) | \(-50577879066661513/621261297432576\) | \(-621261297432576\) | \([3]\) | \(1540\) | \(1.5204\) | \(\Gamma_0(N)\)-optimal |
174.b2 | 174a2 | \([1, 0, 1, 68840, -31810330]\) | \(36079072622241241607/458176313589497856\) | \(-458176313589497856\) | \([]\) | \(4620\) | \(2.0697\) |
Rank
sage: E.rank()
The elliptic curves in class 174a have rank \(0\).
Complex multiplication
The elliptic curves in class 174a do not have complex multiplication.Modular form 174.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.