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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 21840be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.l4 | 21840be1 | \([0, -1, 0, -229496, -42240144]\) | \(326355561310674169/465699780\) | \(1907506298880\) | \([2]\) | \(124416\) | \(1.6278\) | \(\Gamma_0(N)\)-optimal |
| 21840.l5 | 21840be2 | \([0, -1, 0, -227416, -43045520]\) | \(-317562142497484249/12339342574650\) | \(-50541947185766400\) | \([2]\) | \(248832\) | \(1.9744\) | |
| 21840.l3 | 21840be3 | \([0, -1, 0, -292136, -17308560]\) | \(673163386034885929/357608625192000\) | \(1464764928786432000\) | \([2]\) | \(373248\) | \(2.1771\) | |
| 21840.l6 | 21840be4 | \([0, -1, 0, 1113944, -136544144]\) | \(37321015309599759191/23553520979625000\) | \(-96475221932544000000\) | \([2]\) | \(746496\) | \(2.5237\) | |
| 21840.l1 | 21840be5 | \([0, -1, 0, -13636376, 19386011376]\) | \(68463752473882049153689/1817088000000000\) | \(7442792448000000000\) | \([2]\) | \(1119744\) | \(2.7264\) | |
| 21840.l2 | 21840be6 | \([0, -1, 0, -13103896, 20968967920]\) | \(-60752633741424905775769/11197265625000000000\) | \(-45864000000000000000000\) | \([2]\) | \(2239488\) | \(3.0730\) |
Rank
sage: E.rank()
The elliptic curves in class 21840be have rank \(1\).
Complex multiplication
The elliptic curves in class 21840be do not have complex multiplication.Modular form 21840.2.a.be
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
