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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 171600bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 171600.ek4 | 171600bf1 | \([0, 1, 0, 143792, -15042412]\) | \(5137417856375/4510142208\) | \(-288649101312000000\) | \([2]\) | \(1990656\) | \(2.0378\) | \(\Gamma_0(N)\)-optimal |
| 171600.ek3 | 171600bf2 | \([0, 1, 0, -720208, -134274412]\) | \(645532578015625/252306960048\) | \(16147645443072000000\) | \([2]\) | \(3981312\) | \(2.3844\) | |
| 171600.ek2 | 171600bf3 | \([0, 1, 0, -1494208, 938273588]\) | \(-5764706497797625/2612665516032\) | \(-167210593026048000000\) | \([2]\) | \(5971968\) | \(2.5871\) | |
| 171600.ek1 | 171600bf4 | \([0, 1, 0, -26070208, 51220769588]\) | \(30618029936661765625/3678951124992\) | \(235452871999488000000\) | \([2]\) | \(11943936\) | \(2.9337\) |
Rank
sage: E.rank()
The elliptic curves in class 171600bf have rank \(0\).
Complex multiplication
The elliptic curves in class 171600bf do not have complex multiplication.Modular form 171600.2.a.bf
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.
