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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1050k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1050.k7 | 1050k1 | \([1, 1, 1, -12438, 528531]\) | \(13619385906841/6048000\) | \(94500000000\) | \([4]\) | \(2304\) | \(1.0645\) | \(\Gamma_0(N)\)-optimal |
| 1050.k6 | 1050k2 | \([1, 1, 1, -14438, 344531]\) | \(21302308926361/8930250000\) | \(139535156250000\) | \([2, 2]\) | \(4608\) | \(1.4111\) | |
| 1050.k5 | 1050k3 | \([1, 1, 1, -36813, -2081469]\) | \(353108405631241/86318776320\) | \(1348730880000000\) | \([4]\) | \(6912\) | \(1.6138\) | |
| 1050.k4 | 1050k4 | \([1, 1, 1, -108938, -13641469]\) | \(9150443179640281/184570312500\) | \(2883911132812500\) | \([2]\) | \(9216\) | \(1.7577\) | |
| 1050.k8 | 1050k5 | \([1, 1, 1, 48062, 2594531]\) | \(785793873833639/637994920500\) | \(-9968670632812500\) | \([2]\) | \(9216\) | \(1.7577\) | |
| 1050.k2 | 1050k6 | \([1, 1, 1, -548813, -156705469]\) | \(1169975873419524361/108425318400\) | \(1694145600000000\) | \([2, 2]\) | \(13824\) | \(1.9604\) | |
| 1050.k1 | 1050k7 | \([1, 1, 1, -8780813, -10018641469]\) | \(4791901410190533590281/41160000\) | \(643125000000\) | \([2]\) | \(27648\) | \(2.3070\) | |
| 1050.k3 | 1050k8 | \([1, 1, 1, -508813, -180465469]\) | \(-932348627918877961/358766164249920\) | \(-5605721316405000000\) | \([2]\) | \(27648\) | \(2.3070\) |
Rank
sage: E.rank()
The elliptic curves in class 1050k have rank \(0\).
Complex multiplication
The elliptic curves in class 1050k do not have complex multiplication.Modular form 1050.2.a.k
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.
