Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 11774k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11774.m5 | 11774k1 | \([1, 1, 1, -438, 6959]\) | \(-15625/28\) | \(-16655052988\) | \([2]\) | \(8064\) | \(0.65156\) | \(\Gamma_0(N)\)-optimal |
| 11774.m4 | 11774k2 | \([1, 1, 1, -8848, 316447]\) | \(128787625/98\) | \(58292685458\) | \([2]\) | \(16128\) | \(0.99814\) | |
| 11774.m6 | 11774k3 | \([1, 1, 1, 3767, -147785]\) | \(9938375/21952\) | \(-13057561542592\) | \([2]\) | \(24192\) | \(1.2009\) | |
| 11774.m3 | 11774k4 | \([1, 1, 1, -29873, -1641401]\) | \(4956477625/941192\) | \(559842951138632\) | \([2]\) | \(48384\) | \(1.5474\) | |
| 11774.m2 | 11774k5 | \([1, 1, 1, -143408, -21023087]\) | \(-548347731625/1835008\) | \(-1091505552621568\) | \([2]\) | \(72576\) | \(1.7502\) | |
| 11774.m1 | 11774k6 | \([1, 1, 1, -2296368, -1340356975]\) | \(2251439055699625/25088\) | \(14922927477248\) | \([2]\) | \(145152\) | \(2.0967\) |
Rank
sage: E.rank()
The elliptic curves in class 11774k have rank \(1\).
Complex multiplication
The elliptic curves in class 11774k do not have complex multiplication.Modular form 11774.2.a.k
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
