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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 23184bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
23184.bt4 | 23184bz1 | \([0, 0, 0, -86259, -15906382]\) | \(-23771111713777/22848457968\) | \(-68225129917120512\) | \([2]\) | \(184320\) | \(1.9263\) | \(\Gamma_0(N)\)-optimal |
23184.bt3 | 23184bz2 | \([0, 0, 0, -1609779, -785893390]\) | \(154502321244119857/55101928644\) | \(164533477300125696\) | \([2, 2]\) | \(368640\) | \(2.2729\) | |
23184.bt2 | 23184bz3 | \([0, 0, 0, -1841619, -544733422]\) | \(231331938231569617/90942310746882\) | \(271552284813217701888\) | \([2]\) | \(737280\) | \(2.6194\) | |
23184.bt1 | 23184bz4 | \([0, 0, 0, -25754259, -50306221870]\) | \(632678989847546725777/80515134\) | \(240416901881856\) | \([2]\) | \(737280\) | \(2.6194\) |
Rank
sage: E.rank()
The elliptic curves in class 23184bz have rank \(1\).
Complex multiplication
The elliptic curves in class 23184bz do not have complex multiplication.Modular form 23184.2.a.bz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.