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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 54720.eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54720.eu1 | 54720co4 | \([0, 0, 0, -1751052, -891859984]\) | \(3107086841064961/570\) | \(108928696320\) | \([2]\) | \(589824\) | \(1.9535\) | |
| 54720.eu2 | 54720co3 | \([0, 0, 0, -126732, -9239056]\) | \(1177918188481/488703750\) | \(93392741007360000\) | \([4]\) | \(589824\) | \(1.9535\) | |
| 54720.eu3 | 54720co2 | \([0, 0, 0, -109452, -13932304]\) | \(758800078561/324900\) | \(62089356902400\) | \([2, 2]\) | \(294912\) | \(1.6070\) | |
| 54720.eu4 | 54720co1 | \([0, 0, 0, -5772, -288016]\) | \(-111284641/123120\) | \(-23528598405120\) | \([2]\) | \(147456\) | \(1.2604\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54720.eu have rank \(0\).
Complex multiplication
The elliptic curves in class 54720.eu do not have complex multiplication.Modular form 54720.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
