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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 882.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 882.i1 | 882i6 | \([1, -1, 1, -1204160, -508296477]\) | \(2251439055699625/25088\) | \(2151700443648\) | \([2]\) | \(6912\) | \(1.9354\) | |
| 882.i2 | 882i5 | \([1, -1, 1, -75200, -7941405]\) | \(-548347731625/1835008\) | \(-157381518163968\) | \([2]\) | \(3456\) | \(1.5888\) | |
| 882.i3 | 882i4 | \([1, -1, 1, -15665, -614631]\) | \(4956477625/941192\) | \(80722386956232\) | \([2]\) | \(2304\) | \(1.3861\) | |
| 882.i4 | 882i2 | \([1, -1, 1, -4640, 122721]\) | \(128787625/98\) | \(8405079858\) | \([2]\) | \(768\) | \(0.83675\) | |
| 882.i5 | 882i1 | \([1, -1, 1, -230, 2769]\) | \(-15625/28\) | \(-2401451388\) | \([2]\) | \(384\) | \(0.49018\) | \(\Gamma_0(N)\)-optimal |
| 882.i6 | 882i3 | \([1, -1, 1, 1975, -57207]\) | \(9938375/21952\) | \(-1882737888192\) | \([2]\) | \(1152\) | \(1.0395\) |
Rank
sage: E.rank()
The elliptic curves in class 882.i have rank \(0\).
Complex multiplication
The elliptic curves in class 882.i do not have complex multiplication.Modular form 882.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
