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SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 121968.ez
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.ez1 | 121968fv6 | \([0, 0, 0, -78739419, 268928121418]\) | \(10206027697760497/5557167\) | \(29396595420708139008\) | \([2]\) | \(9830400\) | \(3.0642\) | |
121968.ez2 | 121968fv4 | \([0, 0, 0, -4948779, 4152546970]\) | \(2533811507137/58110129\) | \(307394028658515972096\) | \([2, 2]\) | \(4915200\) | \(2.7176\) | |
121968.ez3 | 121968fv2 | \([0, 0, 0, -679899, -120601910]\) | \(6570725617/2614689\) | \(13831319930456641536\) | \([2, 2]\) | \(2457600\) | \(2.3710\) | |
121968.ez4 | 121968fv1 | \([0, 0, 0, -592779, -175609478]\) | \(4354703137/1617\) | \(8553691979255808\) | \([2]\) | \(1228800\) | \(2.0245\) | \(\Gamma_0(N)\)-optimal |
121968.ez5 | 121968fv5 | \([0, 0, 0, 539781, 12858500842]\) | \(3288008303/13504609503\) | \(-71437396406179878531072\) | \([2]\) | \(9830400\) | \(3.0642\) | |
121968.ez6 | 121968fv3 | \([0, 0, 0, 2195061, -873266438]\) | \(221115865823/190238433\) | \(-1006333307667466555392\) | \([4]\) | \(4915200\) | \(2.7176\) |
Rank
sage: E.rank()
The elliptic curves in class 121968.ez have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.ez do not have complex multiplication.Modular form 121968.2.a.ez
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.