Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 18928z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18928.bb5 | 18928z1 | \([0, -1, 0, -1408, -40704]\) | \(-15625/28\) | \(-553577070592\) | \([2]\) | \(17280\) | \(0.94354\) | \(\Gamma_0(N)\)-optimal |
| 18928.bb4 | 18928z2 | \([0, -1, 0, -28448, -1836160]\) | \(128787625/98\) | \(1937519747072\) | \([2]\) | \(34560\) | \(1.2901\) | |
| 18928.bb6 | 18928z3 | \([0, -1, 0, 12112, 857024]\) | \(9938375/21952\) | \(-434004423344128\) | \([2]\) | \(51840\) | \(1.4928\) | |
| 18928.bb3 | 18928z4 | \([0, -1, 0, -96048, 9423296]\) | \(4956477625/941192\) | \(18607939650879488\) | \([2]\) | \(103680\) | \(1.8394\) | |
| 18928.bb2 | 18928z5 | \([0, -1, 0, -461088, 121011968]\) | \(-548347731625/1835008\) | \(-36279226898317312\) | \([2]\) | \(155520\) | \(2.0421\) | |
| 18928.bb1 | 18928z6 | \([0, -1, 0, -7383328, 7724400384]\) | \(2251439055699625/25088\) | \(496005055250432\) | \([2]\) | \(311040\) | \(2.3887\) |
Rank
sage: E.rank()
The elliptic curves in class 18928z have rank \(1\).
Complex multiplication
The elliptic curves in class 18928z do not have complex multiplication.Modular form 18928.2.a.z
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.
