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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 18480.bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18480.bi1 | 18480cc7 | \([0, -1, 0, -4102840, -2999791760]\) | \(1864737106103260904761/129177711985836360\) | \(529111908293985730560\) | \([2]\) | \(663552\) | \(2.7244\) | |
18480.bi2 | 18480cc4 | \([0, -1, 0, -4032040, -3114928400]\) | \(1769857772964702379561/691787250\) | \(2833560576000\) | \([2]\) | \(221184\) | \(2.1751\) | |
18480.bi3 | 18480cc6 | \([0, -1, 0, -810040, 224518000]\) | \(14351050585434661561/3001282273281600\) | \(12293252191361433600\) | \([2, 2]\) | \(331776\) | \(2.3779\) | |
18480.bi4 | 18480cc3 | \([0, -1, 0, -763960, 257253232]\) | \(12038605770121350841/757333463040\) | \(3102037864611840\) | \([2]\) | \(165888\) | \(2.0313\) | |
18480.bi5 | 18480cc2 | \([0, -1, 0, -252040, -48592400]\) | \(432288716775559561/270140062500\) | \(1106493696000000\) | \([2, 2]\) | \(110592\) | \(1.8285\) | |
18480.bi6 | 18480cc5 | \([0, -1, 0, -204520, -67524368]\) | \(-230979395175477481/348191894531250\) | \(-1426194000000000000\) | \([2]\) | \(221184\) | \(2.1751\) | |
18480.bi7 | 18480cc1 | \([0, -1, 0, -18760, -443408]\) | \(178272935636041/81841914000\) | \(335224479744000\) | \([2]\) | \(55296\) | \(1.4820\) | \(\Gamma_0(N)\)-optimal |
18480.bi8 | 18480cc8 | \([0, -1, 0, 1745480, 1353035632]\) | \(143584693754978072519/276341298967965000\) | \(-1131893960572784640000\) | \([2]\) | \(663552\) | \(2.7244\) |
Rank
sage: E.rank()
The elliptic curves in class 18480.bi have rank \(0\).
Complex multiplication
The elliptic curves in class 18480.bi do not have complex multiplication.Modular form 18480.2.a.bi
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.