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SageMath
E = EllipticCurve("ey1")
E.isogeny_class()
Elliptic curves in class 278850ey
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
278850.ey7 | 278850ey1 | \([1, 1, 1, -48249588, -123419094219]\) | \(164711681450297281/8097103872000\) | \(610674591301632000000000\) | \([2]\) | \(55738368\) | \(3.3234\) | \(\Gamma_0(N)\)-optimal |
278850.ey6 | 278850ey2 | \([1, 1, 1, -134777588, 442820137781]\) | \(3590017885052913601/954068544000000\) | \(71954791168689000000000000\) | \([2, 2]\) | \(111476736\) | \(3.6700\) | |
278850.ey3 | 278850ey3 | \([1, 1, 1, -3860889588, -92339337174219]\) | \(84392862605474684114881/11228954880\) | \(846875319927780000000\) | \([2]\) | \(167215104\) | \(3.8727\) | |
278850.ey5 | 278850ey4 | \([1, 1, 1, -1993777588, 34261748137781]\) | \(11621808143080380273601/1335706803288000\) | \(100737525304246062375000000\) | \([2]\) | \(222953472\) | \(4.0165\) | |
278850.ey8 | 278850ey5 | \([1, 1, 1, 339774412, 2864933545781]\) | \(57519563401957999679/80296734375000000\) | \(-6055890627372802734375000000\) | \([2]\) | \(222953472\) | \(4.0165\) | |
278850.ey2 | 278850ey6 | \([1, 1, 1, -3861227588, -92322361462219]\) | \(84415028961834287121601/30783551683856400\) | \(2321661317493800409806250000\) | \([2, 2]\) | \(334430208\) | \(4.2193\) | |
278850.ey1 | 278850ey7 | \([1, 1, 1, -4423575088, -63669631642219]\) | \(126929854754212758768001/50235797102795981820\) | \(3788728087155462034572068437500\) | \([2]\) | \(668860416\) | \(4.5658\) | |
278850.ey4 | 278850ey8 | \([1, 1, 1, -3304288088, -119888638954219]\) | \(-52902632853833942200321/51713453577420277500\) | \(-3900171299196474878429648437500\) | \([2]\) | \(668860416\) | \(4.5658\) |
Rank
sage: E.rank()
The elliptic curves in class 278850ey have rank \(1\).
Complex multiplication
The elliptic curves in class 278850ey do not have complex multiplication.Modular form 278850.2.a.ey
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.