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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 363888bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 363888.bt3 | 363888bt1 | \([0, 0, 0, -97670355, -264807072878]\) | \(19804628171203875/5638671302656\) | \(29337432046704506135642112\) | \([2]\) | \(79626240\) | \(3.5934\) | \(\Gamma_0(N)\)-optimal |
| 363888.bt4 | 363888bt2 | \([0, 0, 0, 257207085, -1746846237806]\) | \(361682234074684125/462672528510976\) | \(-2407238006349844768465354752\) | \([2]\) | \(159252480\) | \(3.9400\) | |
| 363888.bt1 | 363888bt3 | \([0, 0, 0, -7261758675, -238183028307054]\) | \(11165451838341046875/572244736\) | \(2170474323341443662348288\) | \([2]\) | \(238878720\) | \(4.1427\) | |
| 363888.bt2 | 363888bt4 | \([0, 0, 0, -7249282515, -239042229008238]\) | \(-11108001800138902875/79947274872976\) | \(-303233033729322037700041310208\) | \([2]\) | \(477757440\) | \(4.4893\) |
Rank
sage: E.rank()
The elliptic curves in class 363888bt have rank \(1\).
Complex multiplication
The elliptic curves in class 363888bt do not have complex multiplication.Modular form 363888.2.a.bt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
