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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 22050bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.bo7 | 22050bd1 | \([1, -1, 0, -5485167, 4944093741]\) | \(13619385906841/6048000\) | \(8104898434500000000\) | \([2]\) | \(884736\) | \(2.5868\) | \(\Gamma_0(N)\)-optimal |
22050.bo6 | 22050bd2 | \([1, -1, 0, -6367167, 3248007741]\) | \(21302308926361/8930250000\) | \(11967389094691406250000\) | \([2, 2]\) | \(1769472\) | \(2.9334\) | |
22050.bo5 | 22050bd3 | \([1, -1, 0, -16234542, -19130371884]\) | \(353108405631241/86318776320\) | \(115675415850516480000000\) | \([2]\) | \(2654208\) | \(3.1361\) | |
22050.bo8 | 22050bd4 | \([1, -1, 0, 21195333, 23837195241]\) | \(785793873833639/637994920500\) | \(-854974211702943445312500\) | \([2]\) | \(3538944\) | \(3.2799\) | |
22050.bo4 | 22050bd5 | \([1, -1, 0, -48041667, -125901267759]\) | \(9150443179640281/184570312500\) | \(247341871170043945312500\) | \([2]\) | \(3538944\) | \(3.2799\) | |
22050.bo2 | 22050bd6 | \([1, -1, 0, -242026542, -1449071107884]\) | \(1169975873419524361/108425318400\) | \(145300296521217600000000\) | \([2, 2]\) | \(5308416\) | \(3.4827\) | |
22050.bo3 | 22050bd7 | \([1, -1, 0, -224386542, -1669271227884]\) | \(-932348627918877961/358766164249920\) | \(-480780972715070515005000000\) | \([2]\) | \(10616832\) | \(3.8292\) | |
22050.bo1 | 22050bd8 | \([1, -1, 0, -3872338542, -92747787595884]\) | \(4791901410190533590281/41160000\) | \(55158336568125000000\) | \([2]\) | \(10616832\) | \(3.8292\) |
Rank
sage: E.rank()
The elliptic curves in class 22050bd have rank \(1\).
Complex multiplication
The elliptic curves in class 22050bd do not have complex multiplication.Modular form 22050.2.a.bd
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.