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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 274890o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 274890.o7 | 274890o1 | \([1, 1, 0, -5315296928, -370201699642368]\) | \(-141162084764748587904214427641/421539677967044903067648000\) | \(-49593721573144865801005719552000\) | \([2]\) | \(828112896\) | \(4.7686\) | \(\Gamma_0(N)\)-optimal |
| 274890.o6 | 274890o2 | \([1, 1, 0, -116427602848, -15273939739764992]\) | \(1483553933406627878314880715143161/1904972409734563785924000000\) | \(224118099032861694850172676000000\) | \([2, 2]\) | \(1656225792\) | \(5.1152\) | |
| 274890.o8 | 274890o3 | \([1, 1, 0, 46499568097, 8525314183866117]\) | \(94510971880619057444979349412759/321572798571266028122690027520\) | \(-37832718179110876942606359047700480\) | \([2]\) | \(2484338688\) | \(5.3179\) | |
| 274890.o5 | 274890o4 | \([1, 1, 0, -148384912848, -6221411710450992]\) | \(3071176032738522446354893004903161/1635177816170458876705577958000\) | \(192377034894638316385534541180742000\) | \([2]\) | \(3312451584\) | \(5.4618\) | |
| 274890.o3 | 274890o5 | \([1, 1, 0, -1862267187568, -978163480090808528]\) | \(6071016954682394123338855607356153081/10029115297984535156250000\) | \(1179915385692582576597656250000\) | \([2]\) | \(3312451584\) | \(5.4618\) | |
| 274890.o4 | 274890o6 | \([1, 1, 0, -450861000223, 101051190678595333]\) | \(86151626782508161683074667552941161/12360692761105045152384575078400\) | \(1454223142651247457132892873398681600\) | \([2, 2]\) | \(4968677376\) | \(5.6645\) | |
| 274890.o1 | 274890o7 | \([1, 1, 0, -6944332846623, 7043425836680812293]\) | \(314794443646748303921433115102799635561/8206405838866889178408192798720\) | \(965475440536850644950545474576609280\) | \([2]\) | \(9937354752\) | \(6.0111\) | |
| 274890.o2 | 274890o8 | \([1, 1, 0, -1915158246943, -919657412449585403]\) | \(6603124212008881280120689341135103081/715642524575996594697670556160000\) | \(84194627373841423369586243261667840000\) | \([2]\) | \(9937354752\) | \(6.0111\) |
Rank
sage: E.rank()
The elliptic curves in class 274890o have rank \(0\).
Complex multiplication
The elliptic curves in class 274890o do not have complex multiplication.Modular form 274890.2.a.o
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.
