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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 20328m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20328.n4 | 20328m1 | \([0, 1, 0, -251599, -48658810]\) | \(62140690757632/6237\) | \(176787615312\) | \([2]\) | \(92160\) | \(1.5895\) | \(\Gamma_0(N)\)-optimal |
| 20328.n3 | 20328m2 | \([0, 1, 0, -252204, -48413664]\) | \(3911877700432/38900169\) | \(17641989707215104\) | \([2, 2]\) | \(184320\) | \(1.9361\) | |
| 20328.n2 | 20328m3 | \([0, 1, 0, -448224, 37051056]\) | \(5489767279588/2847396321\) | \(5165400344398930944\) | \([2, 2]\) | \(368640\) | \(2.2827\) | |
| 20328.n5 | 20328m4 | \([0, 1, 0, -65864, -118179360]\) | \(-17418812548/3314597517\) | \(-6012939972417573888\) | \([2]\) | \(368640\) | \(2.2827\) | |
| 20328.n1 | 20328m5 | \([0, 1, 0, -5718984, 5257211760]\) | \(5701568801608514/6277868289\) | \(22777091325806856192\) | \([2]\) | \(737280\) | \(2.6293\) | |
| 20328.n6 | 20328m6 | \([0, 1, 0, 1686216, 289768752]\) | \(146142660369886/94532266521\) | \(-342977897697708607488\) | \([2]\) | \(737280\) | \(2.6293\) |
Rank
sage: E.rank()
The elliptic curves in class 20328m have rank \(0\).
Complex multiplication
The elliptic curves in class 20328m do not have complex multiplication.Modular form 20328.2.a.m
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
