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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 281775.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 281775.p1 | 281775p3 | \([1, 1, 1, -188648785313, 31535984488139156]\) | \(1968666709544018637994033129/113621848881699526875\) | \(42852425270149924487697919921875\) | \([2]\) | \(1656225792\) | \(5.1329\) | |
| 281775.p2 | 281775p4 | \([1, 1, 1, -62075816563, -5567198688735844]\) | \(70141892778055497175333129/5090453819946781723125\) | \(1919862192504359690154196611328125\) | \([2]\) | \(1656225792\) | \(5.1329\) | |
| 281775.p3 | 281775p2 | \([1, 1, 1, -12471675938, 432620536420406]\) | \(568832774079017834683129/114800389711906640625\) | \(43296911373406822806939697265625\) | \([2, 2]\) | \(828112896\) | \(4.7863\) | |
| 281775.p4 | 281775p1 | \([1, 1, 1, 1639652187, 40382059857906]\) | \(1292603583867446566871/2615843353271484375\) | \(-986564053637216091156005859375\) | \([2]\) | \(414056448\) | \(4.4398\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 281775.p have rank \(1\).
Complex multiplication
The elliptic curves in class 281775.p do not have complex multiplication.Modular form 281775.2.a.p
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.
