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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 127680.eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127680.eu1 | 127680fe4 | \([0, 1, 0, -5692001, -5046121185]\) | \(77799851782095807001/3092322318750000\) | \(810633741926400000000\) | \([2]\) | \(4718592\) | \(2.7787\) | |
| 127680.eu2 | 127680fe2 | \([0, 1, 0, -925281, 236357919]\) | \(334199035754662681/101099003040000\) | \(26502497052917760000\) | \([2, 2]\) | \(2359296\) | \(2.4321\) | |
| 127680.eu3 | 127680fe1 | \([0, 1, 0, -843361, 297781535]\) | \(253060782505556761/41184460800\) | \(10796259291955200\) | \([2]\) | \(1179648\) | \(2.0855\) | \(\Gamma_0(N)\)-optimal |
| 127680.eu4 | 127680fe3 | \([0, 1, 0, 2530719, 1589036319]\) | \(6837784281928633319/8113766016106800\) | \(-2126975078526300979200\) | \([2]\) | \(4718592\) | \(2.7787\) |
Rank
sage: E.rank()
The elliptic curves in class 127680.eu have rank \(0\).
Complex multiplication
The elliptic curves in class 127680.eu do not have complex multiplication.Modular form 127680.2.a.eu
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
