Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 5175.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5175.q1 | 5175c3 | \([1, -1, 0, -6780042, -6790770509]\) | \(3026030815665395929/1364501953125\) | \(15542530059814453125\) | \([2]\) | \(230400\) | \(2.6403\) | |
| 5175.q2 | 5175c4 | \([1, -1, 0, -3726792, 2721966241]\) | \(502552788401502649/10024505152875\) | \(114185379006966796875\) | \([2]\) | \(230400\) | \(2.6403\) | |
| 5175.q3 | 5175c2 | \([1, -1, 0, -492417, -69299384]\) | \(1159246431432649/488076890625\) | \(5559500832275390625\) | \([2, 2]\) | \(115200\) | \(2.2937\) | |
| 5175.q4 | 5175c1 | \([1, -1, 0, 102708, -8001509]\) | \(10519294081031/8500170375\) | \(-96822253177734375\) | \([2]\) | \(57600\) | \(1.9472\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5175.q have rank \(0\).
Complex multiplication
The elliptic curves in class 5175.q do not have complex multiplication.Modular form 5175.2.a.q
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
