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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 76440.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76440.b1 | 76440bt6 | \([0, -1, 0, -5339056, 4749294796]\) | \(69855246474511682/14613770535\) | \(3521117162848696320\) | \([2]\) | \(2359296\) | \(2.5550\) | |
76440.b2 | 76440bt4 | \([0, -1, 0, -2379456, -1411953444]\) | \(12367124507424964/14926275\) | \(1798206799334400\) | \([2]\) | \(1179648\) | \(2.2084\) | |
76440.b3 | 76440bt3 | \([0, -1, 0, -370456, 56948956]\) | \(46670944188964/15429366225\) | \(1858815495173145600\) | \([2, 2]\) | \(1179648\) | \(2.2084\) | |
76440.b4 | 76440bt2 | \([0, -1, 0, -149956, -21637244]\) | \(12381975627856/419225625\) | \(12626297742240000\) | \([2, 2]\) | \(589824\) | \(1.8618\) | |
76440.b5 | 76440bt1 | \([0, -1, 0, 3169, -1179744]\) | \(1869154304/319921875\) | \(-602215818750000\) | \([2]\) | \(294912\) | \(1.5152\) | \(\Gamma_0(N)\)-optimal |
76440.b6 | 76440bt5 | \([0, -1, 0, 1070144, 390591916]\) | \(562511980386718/599562079935\) | \(-144461576483374725120\) | \([2]\) | \(2359296\) | \(2.5550\) |
Rank
sage: E.rank()
The elliptic curves in class 76440.b have rank \(1\).
Complex multiplication
The elliptic curves in class 76440.b do not have complex multiplication.Modular form 76440.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.