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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 9900.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9900.bd1 | 9900t4 | \([0, 0, 0, -1597575, 777212750]\) | \(154639330142416/33275\) | \(97029900000000\) | \([2]\) | \(124416\) | \(2.0685\) | |
9900.bd2 | 9900t3 | \([0, 0, 0, -100200, 12054125]\) | \(610462990336/8857805\) | \(1614334961250000\) | \([2]\) | \(62208\) | \(1.7219\) | |
9900.bd3 | 9900t2 | \([0, 0, 0, -22575, 737750]\) | \(436334416/171875\) | \(501187500000000\) | \([2]\) | \(41472\) | \(1.5192\) | |
9900.bd4 | 9900t1 | \([0, 0, 0, -10200, -388375]\) | \(643956736/15125\) | \(2756531250000\) | \([2]\) | \(20736\) | \(1.1726\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9900.bd have rank \(0\).
Complex multiplication
The elliptic curves in class 9900.bd do not have complex multiplication.Modular form 9900.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.