Show commands:
SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 26640be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
26640.p5 | 26640be1 | \([0, 0, 0, -121683, 56241682]\) | \(-66730743078481/419010969600\) | \(-1251160051050086400\) | \([2]\) | \(331776\) | \(2.1561\) | \(\Gamma_0(N)\)-optimal |
26640.p4 | 26640be2 | \([0, 0, 0, -3070803, 2066951698]\) | \(1072487167529950801/2554882560000\) | \(7628838446039040000\) | \([2, 2]\) | \(663552\) | \(2.5027\) | |
26640.p3 | 26640be3 | \([0, 0, 0, -4222803, 375124498]\) | \(2788936974993502801/1593609593601600\) | \(4758492748740879974400\) | \([2, 2]\) | \(1327104\) | \(2.8493\) | |
26640.p1 | 26640be4 | \([0, 0, 0, -49104723, 132444219922]\) | \(4385367890843575421521/24975000000\) | \(74574950400000000\) | \([2]\) | \(1327104\) | \(2.8493\) | |
26640.p6 | 26640be5 | \([0, 0, 0, 16772397, 2991126418]\) | \(174751791402194852399/102423900876336360\) | \(-305836129234326349578240\) | \([2]\) | \(2654208\) | \(3.1958\) | |
26640.p2 | 26640be6 | \([0, 0, 0, -43650003, -110517818222]\) | \(3080272010107543650001/15465841417699560\) | \(46180755019788202967040\) | \([2]\) | \(2654208\) | \(3.1958\) |
Rank
sage: E.rank()
The elliptic curves in class 26640be have rank \(0\).
Complex multiplication
The elliptic curves in class 26640be do not have complex multiplication.Modular form 26640.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.