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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1344.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1344.i1 | 1344m3 | \([0, -1, 0, -86017, -9681503]\) | \(268498407453697/252\) | \(66060288\) | \([2]\) | \(3072\) | \(1.2288\) | |
| 1344.i2 | 1344m5 | \([0, -1, 0, -58497, 5412897]\) | \(84448510979617/933897762\) | \(244815694921728\) | \([4]\) | \(6144\) | \(1.5754\) | |
| 1344.i3 | 1344m4 | \([0, -1, 0, -6657, -71775]\) | \(124475734657/63011844\) | \(16518176833536\) | \([2, 2]\) | \(3072\) | \(1.2288\) | |
| 1344.i4 | 1344m2 | \([0, -1, 0, -5377, -149855]\) | \(65597103937/63504\) | \(16647192576\) | \([2, 2]\) | \(1536\) | \(0.88225\) | |
| 1344.i5 | 1344m1 | \([0, -1, 0, -257, -3423]\) | \(-7189057/16128\) | \(-4227858432\) | \([2]\) | \(768\) | \(0.53568\) | \(\Gamma_0(N)\)-optimal |
| 1344.i6 | 1344m6 | \([0, -1, 0, 24703, -579807]\) | \(6359387729183/4218578658\) | \(-1105875083722752\) | \([2]\) | \(6144\) | \(1.5754\) |
Rank
sage: E.rank()
The elliptic curves in class 1344.i have rank \(1\).
Complex multiplication
The elliptic curves in class 1344.i do not have complex multiplication.Modular form 1344.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
