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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 314160.ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 314160.ck1 | 314160ck7 | \([0, -1, 0, -2267537256040, 1314226524737242480]\) | \(314794443646748303921433115102799635561/8206405838866889178408192798720\) | \(33613438315998778074759957703557120\) | \([2]\) | \(4968677376\) | \(5.7313\) | |
| 314160.ck2 | 314160ck8 | \([0, -1, 0, -625357794920, -171597527191386768]\) | \(6603124212008881280120689341135103081/715642524575996594697670556160000\) | \(2931271780663282051881658598031360000\) | \([2]\) | \(4968677376\) | \(5.7313\) | |
| 314160.ck3 | 314160ck5 | \([0, -1, 0, -608087244920, -182514121109946768]\) | \(6071016954682394123338855607356153081/10029115297984535156250000\) | \(41079256260544656000000000000\) | \([2]\) | \(1656225792\) | \(5.1820\) | |
| 314160.ck4 | 314160ck6 | \([0, -1, 0, -147219918440, 18855116788729200]\) | \(86151626782508161683074667552941161/12360692761105045152384575078400\) | \(50629397549486264944167219521126400\) | \([2, 2]\) | \(2484338688\) | \(5.3847\) | |
| 314160.ck5 | 314160ck4 | \([0, -1, 0, -48452216440, -1160818812928400]\) | \(3071176032738522446354893004903161/1635177816170458876705577958000\) | \(6697688335034199558986047315968000\) | \([2]\) | \(1656225792\) | \(5.1820\) | |
| 314160.ck6 | 314160ck2 | \([0, -1, 0, -38017176440, -2849926215680400]\) | \(1483553933406627878314880715143161/1904972409734563785924000000\) | \(7802766990272773267144704000000\) | \([2, 2]\) | \(828112896\) | \(4.8354\) | |
| 314160.ck7 | 314160ck1 | \([0, -1, 0, -1735607160, -69074544017808]\) | \(-141162084764748587904214427641/421539677967044903067648000\) | \(-1726626520953015922965086208000\) | \([2]\) | \(414056448\) | \(4.4888\) | \(\Gamma_0(N)\)-optimal |
| 314160.ck8 | 314160ck3 | \([0, -1, 0, 15183532440, 1590720500860272]\) | \(94510971880619057444979349412759/321572798571266028122690027520\) | \(-1317162182947905651190538352721920\) | \([2]\) | \(1242169344\) | \(5.0381\) |
Rank
sage: E.rank()
The elliptic curves in class 314160.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 314160.ck do not have complex multiplication.Modular form 314160.2.a.ck
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.
