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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 18480bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18480.l8 | 18480bm1 | \([0, -1, 0, 25424, -203840]\) | \(443688652450511/260789760000\) | \(-1068194856960000\) | \([2]\) | \(82944\) | \(1.5731\) | \(\Gamma_0(N)\)-optimal |
18480.l7 | 18480bm2 | \([0, -1, 0, -102576, -1535040]\) | \(29141055407581489/16604321025600\) | \(68011298920857600\) | \([2, 2]\) | \(165888\) | \(1.9197\) | |
18480.l6 | 18480bm3 | \([0, -1, 0, -324016, 78049216]\) | \(-918468938249433649/109183593750000\) | \(-447216000000000000\) | \([2]\) | \(248832\) | \(2.1224\) | |
18480.l4 | 18480bm4 | \([0, -1, 0, -1200176, -504674880]\) | \(46676570542430835889/106752955783320\) | \(437260106888478720\) | \([2]\) | \(331776\) | \(2.2663\) | |
18480.l5 | 18480bm5 | \([0, -1, 0, -1052976, 414360000]\) | \(31522423139920199089/164434491947880\) | \(673523679018516480\) | \([2]\) | \(331776\) | \(2.2663\) | |
18480.l3 | 18480bm6 | \([0, -1, 0, -5324016, 4730049216]\) | \(4074571110566294433649/48828650062500\) | \(200002150656000000\) | \([2, 2]\) | \(497664\) | \(2.4690\) | |
18480.l2 | 18480bm7 | \([0, -1, 0, -5464016, 4468305216]\) | \(4404531606962679693649/444872222400201750\) | \(1822196622951226368000\) | \([2]\) | \(995328\) | \(2.8156\) | |
18480.l1 | 18480bm8 | \([0, -1, 0, -85184016, 302639793216]\) | \(16689299266861680229173649/2396798250\) | \(9817285632000\) | \([2]\) | \(995328\) | \(2.8156\) |
Rank
sage: E.rank()
The elliptic curves in class 18480bm have rank \(1\).
Complex multiplication
The elliptic curves in class 18480bm do not have complex multiplication.Modular form 18480.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.