Show commands:
SageMath
E = EllipticCurve("bv1")
E.isogeny_class()
Elliptic curves in class 22080.bv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22080.bv1 | 22080cr4 | \([0, 1, 0, -7065601, -7231248385]\) | \(148809678420065817601/20700\) | \(5426380800\) | \([2]\) | \(294912\) | \(2.1927\) | |
22080.bv2 | 22080cr6 | \([0, 1, 0, -1653121, 700222079]\) | \(1905890658841300321/293666194803750\) | \(76982830970634240000\) | \([2]\) | \(589824\) | \(2.5393\) | |
22080.bv3 | 22080cr3 | \([0, 1, 0, -453121, -106897921]\) | \(39248884582600321/3935264062500\) | \(1031605862400000000\) | \([2, 2]\) | \(294912\) | \(2.1927\) | |
22080.bv4 | 22080cr2 | \([0, 1, 0, -441601, -113097985]\) | \(36330796409313601/428490000\) | \(112326082560000\) | \([2, 2]\) | \(147456\) | \(1.8462\) | |
22080.bv5 | 22080cr1 | \([0, 1, 0, -26881, -1870081]\) | \(-8194759433281/965779200\) | \(-253173222604800\) | \([2]\) | \(73728\) | \(1.4996\) | \(\Gamma_0(N)\)-optimal |
22080.bv6 | 22080cr5 | \([0, 1, 0, 562559, -517029505]\) | \(75108181893694559/484313964843750\) | \(-126960000000000000000\) | \([2]\) | \(589824\) | \(2.5393\) |
Rank
sage: E.rank()
The elliptic curves in class 22080.bv have rank \(0\).
Complex multiplication
The elliptic curves in class 22080.bv do not have complex multiplication.Modular form 22080.2.a.bv
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.