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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 840.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
840.f1 | 840g5 | \([0, -1, 0, -24400, -1458548]\) | \(784478485879202/221484375\) | \(453600000000\) | \([2]\) | \(2048\) | \(1.2177\) | |
840.f2 | 840g3 | \([0, -1, 0, -1720, -16100]\) | \(549871953124/200930625\) | \(205752960000\) | \([2, 2]\) | \(1024\) | \(0.87118\) | |
840.f3 | 840g2 | \([0, -1, 0, -740, 7812]\) | \(175293437776/4862025\) | \(1244678400\) | \([2, 4]\) | \(512\) | \(0.52460\) | |
840.f4 | 840g1 | \([0, -1, 0, -735, 7920]\) | \(2748251600896/2205\) | \(35280\) | \([4]\) | \(256\) | \(0.17803\) | \(\Gamma_0(N)\)-optimal |
840.f5 | 840g4 | \([0, -1, 0, 160, 24732]\) | \(439608956/259416045\) | \(-265642030080\) | \([4]\) | \(1024\) | \(0.87118\) | |
840.f6 | 840g6 | \([0, -1, 0, 5280, -119700]\) | \(7947184069438/7533176175\) | \(-15427944806400\) | \([2]\) | \(2048\) | \(1.2177\) |
Rank
sage: E.rank()
The elliptic curves in class 840.f have rank \(0\).
Complex multiplication
The elliptic curves in class 840.f do not have complex multiplication.Modular form 840.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.