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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 11858.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11858.bm1 | 11858bk6 | \([1, 1, 1, -16189258, -25078719401]\) | \(2251439055699625/25088\) | \(5228900671672832\) | \([2]\) | \(414720\) | \(2.5850\) | |
| 11858.bm2 | 11858bk5 | \([1, 1, 1, -1011018, -392829865]\) | \(-548347731625/1835008\) | \(-382456734842355712\) | \([2]\) | \(207360\) | \(2.2384\) | |
| 11858.bm3 | 11858bk4 | \([1, 1, 1, -210603, -30579823]\) | \(4956477625/941192\) | \(196165476760726088\) | \([2]\) | \(138240\) | \(2.0357\) | |
| 11858.bm4 | 11858bk2 | \([1, 1, 1, -62378, 5966533]\) | \(128787625/98\) | \(20425393248722\) | \([2]\) | \(46080\) | \(1.4864\) | |
| 11858.bm5 | 11858bk1 | \([1, 1, 1, -3088, 132397]\) | \(-15625/28\) | \(-5835826642492\) | \([2]\) | \(23040\) | \(1.1398\) | \(\Gamma_0(N)\)-optimal |
| 11858.bm6 | 11858bk3 | \([1, 1, 1, 26557, -2784671]\) | \(9938375/21952\) | \(-4575288087713728\) | \([2]\) | \(69120\) | \(1.6891\) |
Rank
sage: E.rank()
The elliptic curves in class 11858.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 11858.bm do not have complex multiplication.Modular form 11858.2.a.bm
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.
