Show commands:
SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 411400bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
411400.bh3 | 411400bh1 | \([0, 0, 0, -432575, -109474750]\) | \(1263257424/425\) | \(3011653700000000\) | \([2]\) | \(2211840\) | \(1.9424\) | \(\Gamma_0(N)\)-optimal* |
411400.bh2 | 411400bh2 | \([0, 0, 0, -493075, -76865250]\) | \(467720676/180625\) | \(5119811290000000000\) | \([2, 2]\) | \(4423680\) | \(2.2890\) | \(\Gamma_0(N)\)-optimal* |
411400.bh1 | 411400bh3 | \([0, 0, 0, -3518075, 2485309750]\) | \(84944038338/2088025\) | \(118370037024800000000\) | \([2]\) | \(8847360\) | \(2.6355\) | \(\Gamma_0(N)\)-optimal* |
411400.bh4 | 411400bh4 | \([0, 0, 0, 1563925, -552032250]\) | \(7462174302/6640625\) | \(-376456712500000000000\) | \([2]\) | \(8847360\) | \(2.6355\) |
Rank
sage: E.rank()
The elliptic curves in class 411400bh have rank \(0\).
Complex multiplication
The elliptic curves in class 411400bh do not have complex multiplication.Modular form 411400.2.a.bh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.