Show commands:
SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 266955w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 266955.w6 | 266955w1 | \([1, 0, 1, -150619, 22485881]\) | \(147281603041/5265\) | \(13508549543385\) | \([2]\) | \(1216512\) | \(1.6086\) | \(\Gamma_0(N)\)-optimal |
| 266955.w5 | 266955w2 | \([1, 0, 1, -157464, 20328337]\) | \(168288035761/27720225\) | \(71122513345922025\) | \([2, 2]\) | \(2433024\) | \(1.9552\) | |
| 266955.w7 | 266955w3 | \([1, 0, 1, 287461, 114474467]\) | \(1023887723039/2798036865\) | \(-7178997077886067785\) | \([2]\) | \(4866048\) | \(2.3017\) | |
| 266955.w4 | 266955w4 | \([1, 0, 1, -711909, -211873229]\) | \(15551989015681/1445900625\) | \(3709785418352105625\) | \([2, 2]\) | \(4866048\) | \(2.3017\) | |
| 266955.w8 | 266955w5 | \([1, 0, 1, 828216, -1002881429]\) | \(24487529386319/183539412225\) | \(-470911917038019950025\) | \([2]\) | \(9732096\) | \(2.6483\) | |
| 266955.w2 | 266955w6 | \([1, 0, 1, -11123154, -14279547473]\) | \(59319456301170001/594140625\) | \(1524402292222265625\) | \([2, 2]\) | \(9732096\) | \(2.6483\) | |
| 266955.w3 | 266955w7 | \([1, 0, 1, -10856199, -14997442859]\) | \(-55150149867714721/5950927734375\) | \(-15268452446136474609375\) | \([2]\) | \(19464192\) | \(2.9949\) | |
| 266955.w1 | 266955w8 | \([1, 0, 1, -177970029, -913851158723]\) | \(242970740812818720001/24375\) | \(62539581219375\) | \([2]\) | \(19464192\) | \(2.9949\) |
Rank
sage: E.rank()
The elliptic curves in class 266955w have rank \(0\).
Complex multiplication
The elliptic curves in class 266955w do not have complex multiplication.Modular form 266955.2.a.w
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
