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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 19110p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19110.e3 | 19110p1 | \([1, 1, 0, -6805617032, -216099982943424]\) | \(296304326013275547793071733369/268420373544960000000\) | \(31579388527190999040000000\) | \([2]\) | \(20643840\) | \(4.1911\) | \(\Gamma_0(N)\)-optimal |
| 19110.e2 | 19110p2 | \([1, 1, 0, -6854789512, -212818772194496]\) | \(302773487204995438715379645049/8911747415025000000000000\) | \(1048458171630276225000000000000\) | \([2, 2]\) | \(41287680\) | \(4.5377\) | |
| 19110.e1 | 19110p3 | \([1, 1, 0, -16216549192, 494082084186496]\) | \(4008766897254067912673785886329/1423480510711669921875000000\) | \(167471058604717254638671875000000\) | \([4]\) | \(82575360\) | \(4.8843\) | |
| 19110.e4 | 19110p4 | \([1, 1, 0, 1720210488, -709721157194496]\) | \(4784981304203817469820354951/1852343836482910078035000000\) | \(-217926400018377887770739715000000\) | \([2]\) | \(82575360\) | \(4.8843\) |
Rank
sage: E.rank()
The elliptic curves in class 19110p have rank \(0\).
Complex multiplication
The elliptic curves in class 19110p do not have complex multiplication.Modular form 19110.2.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 Cremona 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 Cremona labels.
