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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 34272bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34272.k3 | 34272bm1 | \([0, 0, 0, -741, -6820]\) | \(964430272/127449\) | \(5946260544\) | \([2, 2]\) | \(14336\) | \(0.60345\) | \(\Gamma_0(N)\)-optimal |
34272.k4 | 34272bm2 | \([0, 0, 0, 1149, -35926]\) | \(449455096/1753941\) | \(-654654970368\) | \([2]\) | \(28672\) | \(0.95002\) | |
34272.k2 | 34272bm3 | \([0, 0, 0, -3036, 57440]\) | \(1036433728/122451\) | \(365636726784\) | \([4]\) | \(28672\) | \(0.95002\) | |
34272.k1 | 34272bm4 | \([0, 0, 0, -11451, -471634]\) | \(444893916104/9639\) | \(3597737472\) | \([2]\) | \(28672\) | \(0.95002\) |
Rank
sage: E.rank()
The elliptic curves in class 34272bm have rank \(1\).
Complex multiplication
The elliptic curves in class 34272bm do not have complex multiplication.Modular form 34272.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.