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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 2112.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2112.i1 | 2112e3 | \([0, -1, 0, -5153, -140127]\) | \(57736239625/255552\) | \(66991423488\) | \([2]\) | \(2304\) | \(0.92964\) | |
2112.i2 | 2112e4 | \([0, -1, 0, -2593, -281951]\) | \(-7357983625/127552392\) | \(-33437094248448\) | \([2]\) | \(4608\) | \(1.2762\) | |
2112.i3 | 2112e1 | \([0, -1, 0, -353, 2529]\) | \(18609625/1188\) | \(311427072\) | \([2]\) | \(768\) | \(0.38034\) | \(\Gamma_0(N)\)-optimal |
2112.i4 | 2112e2 | \([0, -1, 0, 287, 10081]\) | \(9938375/176418\) | \(-46246920192\) | \([2]\) | \(1536\) | \(0.72691\) |
Rank
sage: E.rank()
The elliptic curves in class 2112.i have rank \(0\).
Complex multiplication
The elliptic curves in class 2112.i do not have complex multiplication.Modular form 2112.2.a.i
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.