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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 103488ff
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
103488.ca3 | 103488ff1 | \([0, -1, 0, -17313, 832833]\) | \(18609625/1188\) | \(36639083593728\) | \([2]\) | \(276480\) | \(1.3533\) | \(\Gamma_0(N)\)-optimal |
103488.ca4 | 103488ff2 | \([0, -1, 0, 14047, 3485889]\) | \(9938375/176418\) | \(-5440903913668608\) | \([2]\) | \(552960\) | \(1.6999\) | |
103488.ca1 | 103488ff3 | \([0, -1, 0, -252513, -48568575]\) | \(57736239625/255552\) | \(7881473981939712\) | \([2]\) | \(829440\) | \(1.9026\) | |
103488.ca2 | 103488ff4 | \([0, -1, 0, -127073, -96963327]\) | \(-7357983625/127552392\) | \(-3933840701235658752\) | \([2]\) | \(1658880\) | \(2.2492\) |
Rank
sage: E.rank()
The elliptic curves in class 103488ff have rank \(1\).
Complex multiplication
The elliptic curves in class 103488ff do not have complex multiplication.Modular form 103488.2.a.ff
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.