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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 145728w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
145728.bx6 | 145728w1 | \([0, 0, 0, 17844, -2689904]\) | \(3288008303/18259263\) | \(-3489399498866688\) | \([2]\) | \(524288\) | \(1.6644\) | \(\Gamma_0(N)\)-optimal |
145728.bx5 | 145728w2 | \([0, 0, 0, -215436, -34695920]\) | \(5786435182177/627352209\) | \(119888874140073984\) | \([2, 2]\) | \(1048576\) | \(2.0110\) | |
145728.bx4 | 145728w3 | \([0, 0, 0, -811596, 244068496]\) | \(309368403125137/44372288367\) | \(8479676358863880192\) | \([2]\) | \(2097152\) | \(2.3576\) | |
145728.bx2 | 145728w4 | \([0, 0, 0, -3351756, -2361845360]\) | \(21790813729717297/304746849\) | \(58238029770522624\) | \([2, 2]\) | \(2097152\) | \(2.3576\) | |
145728.bx3 | 145728w5 | \([0, 0, 0, -3256716, -2502086384]\) | \(-19989223566735457/2584262514273\) | \(-493860257242812776448\) | \([2]\) | \(4194304\) | \(2.7041\) | |
145728.bx1 | 145728w6 | \([0, 0, 0, -53627916, -151159168496]\) | \(89254274298475942657/17457\) | \(3336084652032\) | \([2]\) | \(4194304\) | \(2.7041\) |
Rank
sage: E.rank()
The elliptic curves in class 145728w have rank \(1\).
Complex multiplication
The elliptic curves in class 145728w do not have complex multiplication.Modular form 145728.2.a.w
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.