Show commands:
SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 318402.bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 318402.bn1 | 318402bn3 | \([1, -1, 0, -22239135942, -1276506607549132]\) | \(11165451838341046875/572244736\) | \(62342317789745484724515072\) | \([2]\) | \(477757440\) | \(4.4225\) | |
| 318402.bn2 | 318402bn4 | \([1, -1, 0, -22200927702, -1281111395859100]\) | \(-11108001800138902875/79947274872976\) | \(-8709732222954347757170937525552\) | \([2]\) | \(955514880\) | \(4.7691\) | |
| 318402.bn3 | 318402bn1 | \([1, -1, 0, -299115462, -1419125627340]\) | \(19804628171203875/5638671302656\) | \(842656138394223252527382528\) | \([2]\) | \(159252480\) | \(3.8732\) | \(\Gamma_0(N)\)-optimal |
| 318402.bn4 | 318402bn2 | \([1, -1, 0, 787696698, -9362200979916]\) | \(361682234074684125/462672528510976\) | \(-69142857472913302530561650688\) | \([2]\) | \(318504960\) | \(4.2198\) |
Rank
sage: E.rank()
The elliptic curves in class 318402.bn have rank \(1\).
Complex multiplication
The elliptic curves in class 318402.bn do not have complex multiplication.Modular form 318402.2.a.bn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
