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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 42350.bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42350.bl1 | 42350bb6 | \([1, 1, 0, -8259825, 9133575125]\) | \(2251439055699625/25088\) | \(694451912000000\) | \([2]\) | \(1244160\) | \(2.4168\) | |
42350.bl2 | 42350bb5 | \([1, 1, 0, -515825, 142791125]\) | \(-548347731625/1835008\) | \(-50794196992000000\) | \([2]\) | \(622080\) | \(2.0702\) | |
42350.bl3 | 42350bb4 | \([1, 1, 0, -107450, 11067500]\) | \(4956477625/941192\) | \(26052797511125000\) | \([2]\) | \(414720\) | \(1.8675\) | |
42350.bl4 | 42350bb2 | \([1, 1, 0, -31825, -2197125]\) | \(128787625/98\) | \(2712702781250\) | \([2]\) | \(138240\) | \(1.3182\) | |
42350.bl5 | 42350bb1 | \([1, 1, 0, -1575, -49375]\) | \(-15625/28\) | \(-775057937500\) | \([2]\) | \(69120\) | \(0.97158\) | \(\Gamma_0(N)\)-optimal |
42350.bl6 | 42350bb3 | \([1, 1, 0, 13550, 1024500]\) | \(9938375/21952\) | \(-607645423000000\) | \([2]\) | \(207360\) | \(1.5209\) |
Rank
sage: E.rank()
The elliptic curves in class 42350.bl have rank \(1\).
Complex multiplication
The elliptic curves in class 42350.bl do not have complex multiplication.Modular form 42350.2.a.bl
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.