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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 5040.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.z1 | 5040y3 | \([0, 0, 0, -15147, 716634]\) | \(4767078987/6860\) | \(553063956480\) | \([2]\) | \(6912\) | \(1.1557\) | |
5040.z2 | 5040y4 | \([0, 0, 0, -10827, 1133946]\) | \(-1740992427/5882450\) | \(-474252342681600\) | \([2]\) | \(13824\) | \(1.5022\) | |
5040.z3 | 5040y1 | \([0, 0, 0, -747, -6886]\) | \(416832723/56000\) | \(6193152000\) | \([2]\) | \(2304\) | \(0.60635\) | \(\Gamma_0(N)\)-optimal |
5040.z4 | 5040y2 | \([0, 0, 0, 1173, -36454]\) | \(1613964717/6125000\) | \(-677376000000\) | \([2]\) | \(4608\) | \(0.95293\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.z have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.z do not have complex multiplication.Modular form 5040.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.