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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1584.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1584.i1 | 1584h4 | \([0, 0, 0, -147555, 21816162]\) | \(4406910829875/7744\) | \(624333422592\) | \([2]\) | \(4608\) | \(1.5225\) | |
1584.i2 | 1584h3 | \([0, 0, 0, -9315, 333666]\) | \(1108717875/45056\) | \(3632485367808\) | \([2]\) | \(2304\) | \(1.1759\) | |
1584.i3 | 1584h2 | \([0, 0, 0, -2355, 10994]\) | \(13060888875/7086244\) | \(783681896448\) | \([2]\) | \(1536\) | \(0.97321\) | |
1584.i4 | 1584h1 | \([0, 0, 0, -1395, -19918]\) | \(2714704875/21296\) | \(2355167232\) | \([2]\) | \(768\) | \(0.62663\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1584.i have rank \(0\).
Complex multiplication
The elliptic curves in class 1584.i do not have complex multiplication.Modular form 1584.2.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.