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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 38808.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38808.r1 | 38808bg6 | \([0, 0, 0, -20843571, -36592556434]\) | \(5701568801608514/6277868289\) | \(1102701386335102912512\) | \([2]\) | \(2359296\) | \(2.9526\) | |
| 38808.r2 | 38808bg4 | \([0, 0, 0, -1633611, -258838090]\) | \(5489767279588/2847396321\) | \(250071180699485021184\) | \([2, 2]\) | \(1179648\) | \(2.6060\) | |
| 38808.r3 | 38808bg2 | \([0, 0, 0, -919191, 336273770]\) | \(3911877700432/38900169\) | \(854097049951858944\) | \([2, 2]\) | \(589824\) | \(2.2594\) | |
| 38808.r4 | 38808bg1 | \([0, 0, 0, -916986, 337980881]\) | \(62140690757632/6237\) | \(8558772746832\) | \([2]\) | \(294912\) | \(1.9128\) | \(\Gamma_0(N)\)-optimal |
| 38808.r5 | 38808bg3 | \([0, 0, 0, -240051, 822130526]\) | \(-17418812548/3314597517\) | \(-291102895830345274368\) | \([2]\) | \(1179648\) | \(2.6060\) | |
| 38808.r6 | 38808bg5 | \([0, 0, 0, 6145629, -2012278786]\) | \(146142660369886/94532266521\) | \(-16604499576513198163968\) | \([2]\) | \(2359296\) | \(2.9526\) |
Rank
sage: E.rank()
The elliptic curves in class 38808.r have rank \(0\).
Complex multiplication
The elliptic curves in class 38808.r do not have complex multiplication.Modular form 38808.2.a.r
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.
