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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 38115.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38115.k1 | 38115s6 | \([1, -1, 1, -14429273, -21092093778]\) | \(257260669489908001/14267882475\) | \(18426513201918943275\) | \([2]\) | \(1966080\) | \(2.7621\) | |
| 38115.k2 | 38115s4 | \([1, -1, 1, -952898, -289961328]\) | \(74093292126001/14707625625\) | \(18994427394731105625\) | \([2, 2]\) | \(983040\) | \(2.4155\) | |
| 38115.k3 | 38115s2 | \([1, -1, 1, -294053, 57381756]\) | \(2177286259681/161417025\) | \(208464917438772225\) | \([2, 2]\) | \(491520\) | \(2.0690\) | |
| 38115.k4 | 38115s1 | \([1, -1, 1, -288608, 59749242]\) | \(2058561081361/12705\) | \(16408100546145\) | \([2]\) | \(245760\) | \(1.7224\) | \(\Gamma_0(N)\)-optimal |
| 38115.k5 | 38115s3 | \([1, -1, 1, 277672, 253140396]\) | \(1833318007919/22507682505\) | \(-29067951011629182345\) | \([2]\) | \(983040\) | \(2.4155\) | |
| 38115.k6 | 38115s5 | \([1, -1, 1, 1981957, -1725692394]\) | \(666688497209279/1381398046875\) | \(-1784031329978223046875\) | \([2]\) | \(1966080\) | \(2.7621\) |
Rank
sage: E.rank()
The elliptic curves in class 38115.k have rank \(0\).
Complex multiplication
The elliptic curves in class 38115.k do not have complex multiplication.Modular form 38115.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
