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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 196350cu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 196350.du7 | 196350cu1 | \([1, 1, 1, -2711886188, 134911218784781]\) | \(-141162084764748587904214427641/421539677967044903067648000\) | \(-6586557468235076610432000000000\) | \([4]\) | \(414056448\) | \(4.6004\) | \(\Gamma_0(N)\)-optimal |
| 196350.du6 | 196350cu2 | \([1, 1, 1, -59401838188, 5566262140000781]\) | \(1483553933406627878314880715143161/1904972409734563785924000000\) | \(29765193902102559155062500000000\) | \([2, 2]\) | \(828112896\) | \(4.9470\) | |
| 196350.du8 | 196350cu3 | \([1, 1, 1, 23724269437, -3106875978242719]\) | \(94510971880619057444979349412759/321572798571266028122690027520\) | \(-5024574977676031689417031680000000\) | \([4]\) | \(1242169344\) | \(5.1497\) | |
| 196350.du3 | 196350cu4 | \([1, 1, 1, -950136320188, 356472892792864781]\) | \(6071016954682394123338855607356153081/10029115297984535156250000\) | \(156704926531008361816406250000\) | \([2]\) | \(1656225792\) | \(5.2935\) | |
| 196350.du5 | 196350cu5 | \([1, 1, 1, -75706588188, 2267224244000781]\) | \(3071176032738522446354893004903161/1635177816170458876705577958000\) | \(25549653377663419948524655593750000\) | \([2]\) | \(1656225792\) | \(5.2935\) | |
| 196350.du4 | 196350cu6 | \([1, 1, 1, -230031122563, -36826399977986719]\) | \(86151626782508161683074667552941161/12360692761105045152384575078400\) | \(193135824392266330506008985600000000\) | \([2, 2]\) | \(2484338688\) | \(5.4963\) | |
| 196350.du2 | 196350cu7 | \([1, 1, 1, -977121554563, 335151420295677281]\) | \(6603124212008881280120689341135103081/715642524575996594697670556160000\) | \(11181914446499946792151102440000000000\) | \([2]\) | \(4968677376\) | \(5.8428\) | |
| 196350.du1 | 196350cu8 | \([1, 1, 1, -3543026962563, -2566848681127426719]\) | \(314794443646748303921433115102799635561/8206405838866889178408192798720\) | \(128225091232295143412628012480000000\) | \([2]\) | \(4968677376\) | \(5.8428\) |
Rank
sage: E.rank()
The elliptic curves in class 196350cu have rank \(0\).
Complex multiplication
The elliptic curves in class 196350cu do not have complex multiplication.Modular form 196350.2.a.cu
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.
