Show commands:
SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 101430bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
101430.v4 | 101430bg1 | \([1, -1, 0, -503190, 411463156]\) | \(-164287467238609/757170892800\) | \(-64939610409562828800\) | \([2]\) | \(3538944\) | \(2.4864\) | \(\Gamma_0(N)\)-optimal |
101430.v3 | 101430bg2 | \([1, -1, 0, -11933910, 15845221300]\) | \(2191574502231419089/4115217960000\) | \(352946281498733160000\) | \([2, 2]\) | \(7077888\) | \(2.8330\) | |
101430.v2 | 101430bg3 | \([1, -1, 0, -15902910, 4401006700]\) | \(5186062692284555089/2903809817953800\) | \(249048504207613583209800\) | \([2]\) | \(14155776\) | \(3.1795\) | |
101430.v1 | 101430bg4 | \([1, -1, 0, -190856430, 1014912788476]\) | \(8964546681033941529169/31696875000\) | \(2718518016571875000\) | \([2]\) | \(14155776\) | \(3.1795\) |
Rank
sage: E.rank()
The elliptic curves in class 101430bg have rank \(0\).
Complex multiplication
The elliptic curves in class 101430bg do not have complex multiplication.Modular form 101430.2.a.bg
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.