Show commands:
SageMath
E = EllipticCurve("eo1")
E.isogeny_class()
Elliptic curves in class 163170eo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163170.w6 | 163170eo1 | \([1, -1, 0, 4056750, 6043774036]\) | \(86087999924407151/233867044454400\) | \(-20057869232588449382400\) | \([2]\) | \(11943936\) | \(2.9630\) | \(\Gamma_0(N)\)-optimal |
| 163170.w5 | 163170eo2 | \([1, -1, 0, -35456850, 68688635476]\) | \(57478893731908015249/9504385397982720\) | \(815154268074019119429120\) | \([2]\) | \(23887872\) | \(3.3096\) | |
| 163170.w4 | 163170eo3 | \([1, -1, 0, -37855890, -199943134700]\) | \(-69953320343800203409/160900419519000000\) | \(-13799804849417315799000000\) | \([2]\) | \(35831808\) | \(3.5123\) | |
| 163170.w3 | 163170eo4 | \([1, -1, 0, -794170890, -8606686885700]\) | \(645877025383384520763409/641552191391223000\) | \(55023442874674790155983000\) | \([2]\) | \(71663616\) | \(3.8589\) | |
| 163170.w2 | 163170eo5 | \([1, -1, 0, -4019530230, -98085991178624]\) | \(-83740170636734921311132369/82992553710937500\) | \(-7117949403671264648437500\) | \([2]\) | \(107495424\) | \(4.0616\) | |
| 163170.w1 | 163170eo6 | \([1, -1, 0, -64312498980, -6277548534147374]\) | \(342999983683000258740998632369/336902343750\) | \(28894807179246093750\) | \([2]\) | \(214990848\) | \(4.4082\) |
Rank
sage: E.rank()
The elliptic curves in class 163170eo have rank \(0\).
Complex multiplication
The elliptic curves in class 163170eo do not have complex multiplication.Modular form 163170.2.a.eo
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
