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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 135240.bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
135240.bh1 | 135240bf3 | \([0, -1, 0, -105848640, 419191613100]\) | \(544328872410114151778/14166950625\) | \(3413458071717120000\) | \([2]\) | \(9437184\) | \(3.0715\) | |
135240.bh2 | 135240bf4 | \([0, -1, 0, -10275120, -1467900468]\) | \(497927680189263938/284271240234375\) | \(68493777187500000000000\) | \([2]\) | \(9437184\) | \(3.0715\) | |
135240.bh3 | 135240bf2 | \([0, -1, 0, -6623640, 6534683100]\) | \(266763091319403556/1355769140625\) | \(163333000832400000000\) | \([2, 2]\) | \(4718592\) | \(2.7250\) | |
135240.bh4 | 135240bf1 | \([0, -1, 0, -193860, 210351492]\) | \(-26752376766544/618796614375\) | \(-18637005538458720000\) | \([2]\) | \(2359296\) | \(2.3784\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 135240.bh have rank \(0\).
Complex multiplication
The elliptic curves in class 135240.bh do not have complex multiplication.Modular form 135240.2.a.bh
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.