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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 7728.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7728.b1 | 7728l4 | \([0, -1, 0, -2861584, 1864147264]\) | \(632678989847546725777/80515134\) | \(329789988864\) | \([4]\) | \(92160\) | \(2.0701\) | |
7728.b2 | 7728l3 | \([0, -1, 0, -204624, 20243520]\) | \(231331938231569617/90942310746882\) | \(372499704819228672\) | \([2]\) | \(92160\) | \(2.0701\) | |
7728.b3 | 7728l2 | \([0, -1, 0, -178864, 29166784]\) | \(154502321244119857/55101928644\) | \(225697499725824\) | \([2, 2]\) | \(46080\) | \(1.7236\) | |
7728.b4 | 7728l1 | \([0, -1, 0, -9584, 592320]\) | \(-23771111713777/22848457968\) | \(-93587283836928\) | \([2]\) | \(23040\) | \(1.3770\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7728.b have rank \(1\).
Complex multiplication
The elliptic curves in class 7728.b do not have complex multiplication.Modular form 7728.2.a.b
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.