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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 22080.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22080.bd1 | 22080m4 | \([0, -1, 0, -8640705, -9773360703]\) | \(544328872410114151778/14166950625\) | \(1856890552320000\) | \([2]\) | \(393216\) | \(2.4451\) | |
| 22080.bd2 | 22080m3 | \([0, -1, 0, -838785, 34596225]\) | \(497927680189263938/284271240234375\) | \(37260000000000000000\) | \([4]\) | \(393216\) | \(2.4451\) | |
| 22080.bd3 | 22080m2 | \([0, -1, 0, -540705, -152180703]\) | \(266763091319403556/1355769140625\) | \(88851686400000000\) | \([2, 2]\) | \(196608\) | \(2.0986\) | |
| 22080.bd4 | 22080m1 | \([0, -1, 0, -15825, -4899375]\) | \(-26752376766544/618796614375\) | \(-10138363729920000\) | \([2]\) | \(98304\) | \(1.7520\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 22080.bd have rank \(0\).
Complex multiplication
The elliptic curves in class 22080.bd do not have complex multiplication.Modular form 22080.2.a.bd
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.
