Show commands:
SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 24255bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 24255.bn4 | 24255bm1 | \([1, -1, 0, -116874, 15408063]\) | \(2058561081361/12705\) | \(1089658567305\) | \([2]\) | \(98304\) | \(1.4964\) | \(\Gamma_0(N)\)-optimal |
| 24255.bn3 | 24255bm2 | \([1, -1, 0, -119079, 14798160]\) | \(2177286259681/161417025\) | \(13844112097610025\) | \([2, 2]\) | \(196608\) | \(1.8430\) | |
| 24255.bn5 | 24255bm3 | \([1, -1, 0, 112446, 65224305]\) | \(1833318007919/22507682505\) | \(-1930396621153413105\) | \([2]\) | \(393216\) | \(2.1896\) | |
| 24255.bn2 | 24255bm4 | \([1, -1, 0, -385884, -74688237]\) | \(74093292126001/14707625625\) | \(1261415998976450625\) | \([2, 2]\) | \(393216\) | \(2.1896\) | |
| 24255.bn6 | 24255bm5 | \([1, -1, 0, 802611, -444785580]\) | \(666688497209279/1381398046875\) | \(-118477152037444921875\) | \([2]\) | \(786432\) | \(2.5361\) | |
| 24255.bn1 | 24255bm6 | \([1, -1, 0, -5843259, -5434921962]\) | \(257260669489908001/14267882475\) | \(1223700934764629475\) | \([2]\) | \(786432\) | \(2.5361\) |
Rank
sage: E.rank()
The elliptic curves in class 24255bm have rank \(0\).
Complex multiplication
The elliptic curves in class 24255bm do not have complex multiplication.Modular form 24255.2.a.bm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
