Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 175b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
175.b2 | 175b1 | \([0, -1, 1, -33, 93]\) | \(-262144/35\) | \(-546875\) | \([]\) | \(16\) | \(-0.16643\) | \(\Gamma_0(N)\)-optimal |
175.b3 | 175b2 | \([0, -1, 1, 217, -282]\) | \(71991296/42875\) | \(-669921875\) | \([]\) | \(48\) | \(0.38287\) | |
175.b1 | 175b3 | \([0, -1, 1, -3283, -74657]\) | \(-250523582464/13671875\) | \(-213623046875\) | \([]\) | \(144\) | \(0.93218\) |
Rank
sage: E.rank()
The elliptic curves in class 175b have rank \(1\).
Complex multiplication
The elliptic curves in class 175b do not have complex multiplication.Modular form 175.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.