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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 31920bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31920.ba4 | 31920bk1 | \([0, -1, 0, -161680, -24968768]\) | \(114113060120923921/124104960\) | \(508333916160\) | \([2]\) | \(184320\) | \(1.5336\) | \(\Gamma_0(N)\)-optimal |
| 31920.ba3 | 31920bk2 | \([0, -1, 0, -162960, -24552000]\) | \(116844823575501841/3760263939600\) | \(15402041096601600\) | \([2, 2]\) | \(368640\) | \(1.8802\) | |
| 31920.ba5 | 31920bk3 | \([0, -1, 0, 49840, -84306240]\) | \(3342636501165359/751262567039460\) | \(-3077171474593628160\) | \([2]\) | \(737280\) | \(2.2268\) | |
| 31920.ba2 | 31920bk4 | \([0, -1, 0, -396240, 61854912]\) | \(1679731262160129361/570261564022500\) | \(2335791366236160000\) | \([2, 4]\) | \(737280\) | \(2.2268\) | |
| 31920.ba6 | 31920bk5 | \([0, -1, 0, 1163280, 427406400]\) | \(42502666283088696719/43898058864843750\) | \(-179806449110400000000\) | \([4]\) | \(1474560\) | \(2.5733\) | |
| 31920.ba1 | 31920bk6 | \([0, -1, 0, -5688240, 5222613312]\) | \(4969327007303723277361/1123462695162150\) | \(4601703199384166400\) | \([4]\) | \(1474560\) | \(2.5733\) |
Rank
sage: E.rank()
The elliptic curves in class 31920bk have rank \(1\).
Complex multiplication
The elliptic curves in class 31920bk do not have complex multiplication.Modular form 31920.2.a.bk
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
