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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 26928bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
26928.l3 | 26928bp1 | \([0, 0, 0, -55731, 5063794]\) | \(6411014266033/296208\) | \(884472348672\) | \([2]\) | \(73728\) | \(1.3676\) | \(\Gamma_0(N)\)-optimal |
26928.l2 | 26928bp2 | \([0, 0, 0, -58611, 4511410]\) | \(7457162887153/1370924676\) | \(4093559147741184\) | \([2, 2]\) | \(147456\) | \(1.7141\) | |
26928.l4 | 26928bp3 | \([0, 0, 0, 115629, 26221714]\) | \(57258048889007/132611470002\) | \(-395975727642451968\) | \([2]\) | \(294912\) | \(2.0607\) | |
26928.l1 | 26928bp4 | \([0, 0, 0, -278931, -52551470]\) | \(803760366578833/65593817586\) | \(195862089810714624\) | \([2]\) | \(294912\) | \(2.0607\) |
Rank
sage: E.rank()
The elliptic curves in class 26928bp have rank \(2\).
Complex multiplication
The elliptic curves in class 26928bp do not have complex multiplication.Modular form 26928.2.a.bp
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.