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SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 216384ez
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
216384.fj4 | 216384ez1 | \([0, 1, 0, -1878529, 1615933535]\) | \(-23771111713777/22848457968\) | \(-704668822792367505408\) | \([2]\) | \(8847360\) | \(2.6965\) | \(\Gamma_0(N)\)-optimal |
216384.fj3 | 216384ez2 | \([0, 1, 0, -35057409, 79858368351]\) | \(154502321244119857/55101928644\) | \(1699397449295581937664\) | \([2, 2]\) | \(17694720\) | \(3.0431\) | |
216384.fj1 | 216384ez3 | \([0, 1, 0, -560870529, 5112415739871]\) | \(632678989847546725777/80515134\) | \(2483165593591087104\) | \([2]\) | \(35389440\) | \(3.3897\) | |
216384.fj2 | 216384ez4 | \([0, 1, 0, -40106369, 55347687135]\) | \(231331938231569617/90942310746882\) | \(2804749937425755778056192\) | \([2]\) | \(35389440\) | \(3.3897\) |
Rank
sage: E.rank()
The elliptic curves in class 216384ez have rank \(2\).
Complex multiplication
The elliptic curves in class 216384ez do not have complex multiplication.Modular form 216384.2.a.ez
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.