Show commands:
SageMath
E = EllipticCurve("gm1")
E.isogeny_class()
Elliptic curves in class 172480gm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
172480.fy2 | 172480gm1 | \([0, 1, 0, 495, -3025]\) | \(16674224/15125\) | \(-12142592000\) | \([]\) | \(110592\) | \(0.62234\) | \(\Gamma_0(N)\)-optimal |
172480.fy1 | 172480gm2 | \([0, 1, 0, -5105, 188495]\) | \(-18330740176/8857805\) | \(-7111187578880\) | \([]\) | \(331776\) | \(1.1716\) |
Rank
sage: E.rank()
The elliptic curves in class 172480gm have rank \(2\).
Complex multiplication
The elliptic curves in class 172480gm do not have complex multiplication.Modular form 172480.2.a.gm
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.