Show commands:
SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 53040.bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53040.bb1 | 53040m6 | \([0, -1, 0, -194480, 32821152]\) | \(397210600760070242/3536192675535\) | \(7242122599495680\) | \([4]\) | \(360448\) | \(1.8664\) | |
| 53040.bb2 | 53040m4 | \([0, -1, 0, -21080, -332928]\) | \(1011710313226084/536724738225\) | \(549606131942400\) | \([2, 4]\) | \(180224\) | \(1.5198\) | |
| 53040.bb3 | 53040m2 | \([0, -1, 0, -16580, -815328]\) | \(1969080716416336/2472575625\) | \(632979360000\) | \([2, 2]\) | \(90112\) | \(1.1732\) | |
| 53040.bb4 | 53040m1 | \([0, -1, 0, -16575, -815850]\) | \(31476797652269056/49725\) | \(795600\) | \([2]\) | \(45056\) | \(0.82665\) | \(\Gamma_0(N)\)-optimal |
| 53040.bb5 | 53040m3 | \([0, -1, 0, -12160, -1264400]\) | \(-194204905090564/566398828125\) | \(-579992400000000\) | \([2]\) | \(180224\) | \(1.5198\) | |
| 53040.bb6 | 53040m5 | \([0, -1, 0, 80320, -2685408]\) | \(27980756504588158/17683545112935\) | \(-36215900391290880\) | \([4]\) | \(360448\) | \(1.8664\) |
Rank
sage: E.rank()
The elliptic curves in class 53040.bb have rank \(0\).
Complex multiplication
The elliptic curves in class 53040.bb do not have complex multiplication.Modular form 53040.2.a.bb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
