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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 60690.bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60690.bo1 | 60690bm7 | \([1, 1, 1, -3843661280, -91653993754675]\) | \(260174968233082037895439009/223081361502731896500\) | \(5384641755886134840269608500\) | \([2]\) | \(63700992\) | \(4.2478\) | |
| 60690.bo2 | 60690bm8 | \([1, 1, 1, -2524376280, 48299495237325]\) | \(73704237235978088924479009/899277423164136103500\) | \(21706370851766533523622391500\) | \([2]\) | \(63700992\) | \(4.2478\) | |
| 60690.bo3 | 60690bm5 | \([1, 1, 1, -2516937420, 48601242116397]\) | \(73054578035931991395831649/136386452160\) | \(3292037399677199040\) | \([2]\) | \(21233664\) | \(3.6985\) | |
| 60690.bo4 | 60690bm6 | \([1, 1, 1, -294018780, -743389758675]\) | \(116454264690812369959009/57505157319440250000\) | \(1388034702653844075752250000\) | \([2, 2]\) | \(31850496\) | \(3.9012\) | |
| 60690.bo5 | 60690bm4 | \([1, 1, 1, -165171020, 679236591917]\) | \(20645800966247918737249/3688936444974392640\) | \(89041957977184105581092160\) | \([2]\) | \(21233664\) | \(3.6985\) | |
| 60690.bo6 | 60690bm2 | \([1, 1, 1, -157310220, 759328710957]\) | \(17836145204788591940449/770635366502400\) | \(18601264332791968665600\) | \([2, 2]\) | \(10616832\) | \(3.3519\) | |
| 60690.bo7 | 60690bm1 | \([1, 1, 1, -9342220, 13096493357]\) | \(-3735772816268612449/909650165760000\) | \(-21956743641893437440000\) | \([4]\) | \(5308416\) | \(3.0054\) | \(\Gamma_0(N)\)-optimal |
| 60690.bo8 | 60690bm3 | \([1, 1, 1, 67231220, -89093758675]\) | \(1392333139184610040991/947901937500000000\) | \(-22880048421639937500000000\) | \([4]\) | \(15925248\) | \(3.5547\) |
Rank
sage: E.rank()
The elliptic curves in class 60690.bo have rank \(1\).
Complex multiplication
The elliptic curves in class 60690.bo do not have complex multiplication.Modular form 60690.2.a.bo
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
