Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 210.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
210.b1 | 210b7 | \([1, 0, 1, -351233, -80149132]\) | \(4791901410190533590281/41160000\) | \(41160000\) | \([2]\) | \(1152\) | \(1.5023\) | |
210.b2 | 210b6 | \([1, 0, 1, -21953, -1253644]\) | \(1169975873419524361/108425318400\) | \(108425318400\) | \([2, 2]\) | \(576\) | \(1.1557\) | |
210.b3 | 210b8 | \([1, 0, 1, -20353, -1443724]\) | \(-932348627918877961/358766164249920\) | \(-358766164249920\) | \([4]\) | \(1152\) | \(1.5023\) | |
210.b4 | 210b4 | \([1, 0, 1, -4358, -109132]\) | \(9150443179640281/184570312500\) | \(184570312500\) | \([6]\) | \(384\) | \(0.95296\) | |
210.b5 | 210b3 | \([1, 0, 1, -1473, -16652]\) | \(353108405631241/86318776320\) | \(86318776320\) | \([2]\) | \(288\) | \(0.80912\) | |
210.b6 | 210b2 | \([1, 0, 1, -578, 2756]\) | \(21302308926361/8930250000\) | \(8930250000\) | \([2, 6]\) | \(192\) | \(0.60639\) | |
210.b7 | 210b1 | \([1, 0, 1, -498, 4228]\) | \(13619385906841/6048000\) | \(6048000\) | \([6]\) | \(96\) | \(0.25981\) | \(\Gamma_0(N)\)-optimal |
210.b8 | 210b5 | \([1, 0, 1, 1922, 20756]\) | \(785793873833639/637994920500\) | \(-637994920500\) | \([12]\) | \(384\) | \(0.95296\) |
Rank
sage: E.rank()
The elliptic curves in class 210.b have rank \(0\).
Complex multiplication
The elliptic curves in class 210.b do not have complex multiplication.Modular form 210.2.a.b
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.