Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1225.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1225.b1 | 1225h2 | \([1, 1, 1, -208083, -36621194]\) | \(-162677523113838677\) | \(-6125\) | \([]\) | \(1776\) | \(1.2705\) | |
1225.b2 | 1225h1 | \([1, 1, 1, -8, 6]\) | \(-9317\) | \(-6125\) | \([]\) | \(48\) | \(-0.53497\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1225.b have rank \(1\).
Complex multiplication
The elliptic curves in class 1225.b do not have complex multiplication.Modular form 1225.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 37 \\ 37 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.