Show commands:
SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 317680bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 317680.bo3 | 317680bo1 | \([0, 0, 0, -5674395467, -164523338435526]\) | \(104857852278310619039721/47155625\) | \(9086885569456640000\) | \([2]\) | \(101744640\) | \(3.8812\) | \(\Gamma_0(N)\)-optimal |
| 317680.bo2 | 317680bo2 | \([0, 0, 0, -5674424347, -164521580007414]\) | \(104859453317683374662841/2223652969140625\) | \(428497768331208769600000000\) | \([2, 2]\) | \(203489280\) | \(4.2278\) | |
| 317680.bo1 | 317680bo3 | \([0, 0, 0, -5872974347, -152390135297414]\) | \(116256292809537371612841/15216540068579856875\) | \(2932226184389180561564802560000\) | \([2]\) | \(406978560\) | \(4.5743\) | |
| 317680.bo4 | 317680bo4 | \([0, 0, 0, -5476336427, -176540485318246]\) | \(-94256762600623910012361/15323275604248046875\) | \(-2952794114488961875000000000000\) | \([4]\) | \(406978560\) | \(4.5743\) |
Rank
sage: E.rank()
The elliptic curves in class 317680bo have rank \(1\).
Complex multiplication
The elliptic curves in class 317680bo do not have complex multiplication.Modular form 317680.2.a.bo
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.
