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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 26520bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 26520.bc4 | 26520bf1 | \([0, 1, 0, -16575, 815850]\) | \(31476797652269056/49725\) | \(795600\) | \([4]\) | \(22528\) | \(0.82665\) | \(\Gamma_0(N)\)-optimal |
| 26520.bc3 | 26520bf2 | \([0, 1, 0, -16580, 815328]\) | \(1969080716416336/2472575625\) | \(632979360000\) | \([2, 4]\) | \(45056\) | \(1.1732\) | |
| 26520.bc5 | 26520bf3 | \([0, 1, 0, -12160, 1264400]\) | \(-194204905090564/566398828125\) | \(-579992400000000\) | \([4]\) | \(90112\) | \(1.5198\) | |
| 26520.bc2 | 26520bf4 | \([0, 1, 0, -21080, 332928]\) | \(1011710313226084/536724738225\) | \(549606131942400\) | \([2, 2]\) | \(90112\) | \(1.5198\) | |
| 26520.bc6 | 26520bf5 | \([0, 1, 0, 80320, 2685408]\) | \(27980756504588158/17683545112935\) | \(-36215900391290880\) | \([2]\) | \(180224\) | \(1.8664\) | |
| 26520.bc1 | 26520bf6 | \([0, 1, 0, -194480, -32821152]\) | \(397210600760070242/3536192675535\) | \(7242122599495680\) | \([2]\) | \(180224\) | \(1.8664\) |
Rank
sage: E.rank()
The elliptic curves in class 26520bf have rank \(0\).
Complex multiplication
The elliptic curves in class 26520bf do not have complex multiplication.Modular form 26520.2.a.bf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
