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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 9360bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9360.o6 | 9360bm1 | \([0, 0, 0, -15843, 767522]\) | \(147281603041/5265\) | \(15721205760\) | \([2]\) | \(12288\) | \(1.0456\) | \(\Gamma_0(N)\)-optimal |
| 9360.o5 | 9360bm2 | \([0, 0, 0, -16563, 693938]\) | \(168288035761/27720225\) | \(82772148326400\) | \([2, 2]\) | \(24576\) | \(1.3921\) | |
| 9360.o4 | 9360bm3 | \([0, 0, 0, -74883, -7225918]\) | \(15551989015681/1445900625\) | \(4317436131840000\) | \([2, 2]\) | \(49152\) | \(1.7387\) | |
| 9360.o7 | 9360bm4 | \([0, 0, 0, 30237, 3904418]\) | \(1023887723039/2798036865\) | \(-8354893310300160\) | \([2]\) | \(49152\) | \(1.7387\) | |
| 9360.o2 | 9360bm5 | \([0, 0, 0, -1170003, -487107502]\) | \(59319456301170001/594140625\) | \(1774094400000000\) | \([2, 2]\) | \(98304\) | \(2.0853\) | |
| 9360.o8 | 9360bm6 | \([0, 0, 0, 87117, -34215118]\) | \(24487529386319/183539412225\) | \(-548045748273254400\) | \([2]\) | \(98304\) | \(2.0853\) | |
| 9360.o1 | 9360bm7 | \([0, 0, 0, -18720003, -31175037502]\) | \(242970740812818720001/24375\) | \(72783360000\) | \([2]\) | \(196608\) | \(2.4319\) | |
| 9360.o3 | 9360bm8 | \([0, 0, 0, -1141923, -511598878]\) | \(-55150149867714721/5950927734375\) | \(-17769375000000000000\) | \([2]\) | \(196608\) | \(2.4319\) |
Rank
sage: E.rank()
The elliptic curves in class 9360bm have rank \(0\).
Complex multiplication
The elliptic curves in class 9360bm do not have complex multiplication.Modular form 9360.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
