Show commands:
SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 164934bq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
164934.dq3 | 164934bq1 | \([1, -1, 1, -3651269, -2415404995]\) | \(62768149033310713/6915442583808\) | \(593110685411429568768\) | \([2]\) | \(8847360\) | \(2.7194\) | \(\Gamma_0(N)\)-optimal |
164934.dq2 | 164934bq2 | \([1, -1, 1, -13847189, 17246407133]\) | \(3423676911662954233/483711578981136\) | \(41486065811997166893456\) | \([2, 2]\) | \(17694720\) | \(3.0660\) | |
164934.dq1 | 164934bq3 | \([1, -1, 1, -213417329, 1200058712885]\) | \(12534210458299016895673/315581882565708\) | \(27066233925538302778668\) | \([2]\) | \(35389440\) | \(3.4125\) | |
164934.dq4 | 164934bq4 | \([1, -1, 1, 22588231, 92667726533]\) | \(14861225463775641287/51859390496937804\) | \(-4447778760346617827338284\) | \([2]\) | \(35389440\) | \(3.4125\) |
Rank
sage: E.rank()
The elliptic curves in class 164934bq have rank \(0\).
Complex multiplication
The elliptic curves in class 164934bq do not have complex multiplication.Modular form 164934.2.a.bq
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.