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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 177870gp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.dg6 | 177870gp1 | \([1, 0, 1, 1511771, 68120336]\) | \(1833318007919/1070530560\) | \(-223122527273210019840\) | \([2]\) | \(8847360\) | \(2.5939\) | \(\Gamma_0(N)\)-optimal |
| 177870.dg5 | 177870gp2 | \([1, 0, 1, -6077349, 544717072]\) | \(119102750067601/68309049600\) | \(14237134699249553414400\) | \([2, 2]\) | \(17694720\) | \(2.9405\) | |
| 177870.dg2 | 177870gp3 | \([1, 0, 1, -70110549, 225454928752]\) | \(182864522286982801/463015182960\) | \(96502726449866996083440\) | \([2]\) | \(35389440\) | \(3.2871\) | |
| 177870.dg3 | 177870gp4 | \([1, 0, 1, -63470069, -193832947024]\) | \(135670761487282321/643043610000\) | \(134024679697222681290000\) | \([2, 2]\) | \(35389440\) | \(3.2871\) | |
| 177870.dg4 | 177870gp5 | \([1, 0, 1, -30860569, -392737853224]\) | \(-15595206456730321/310672490129100\) | \(-64751099820883380879489900\) | \([2]\) | \(70778880\) | \(3.6337\) | |
| 177870.dg1 | 177870gp6 | \([1, 0, 1, -1014363089, -12434868972088]\) | \(553808571467029327441/12529687500\) | \(2611467290521392187500\) | \([2]\) | \(70778880\) | \(3.6337\) |
Rank
sage: E.rank()
The elliptic curves in class 177870gp have rank \(0\).
Complex multiplication
The elliptic curves in class 177870gp do not have complex multiplication.Modular form 177870.2.a.gp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
