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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 14994z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14994.b3 | 14994z1 | \([1, -1, 0, 47619, 107925061]\) | \(139233463487/58763045376\) | \(-5039878460046506496\) | \([]\) | \(414720\) | \(2.2678\) | \(\Gamma_0(N)\)-optimal |
14994.b2 | 14994z2 | \([1, -1, 0, -428661, -2917691267]\) | \(-101566487155393/42823570577256\) | \(-3672811535780977943976\) | \([]\) | \(1244160\) | \(2.8171\) | |
14994.b1 | 14994z3 | \([1, -1, 0, -168330591, -840593126129]\) | \(-6150311179917589675873/244053849830826\) | \(-20931552015106452245946\) | \([]\) | \(3732480\) | \(3.3664\) |
Rank
sage: E.rank()
The elliptic curves in class 14994z have rank \(1\).
Complex multiplication
The elliptic curves in class 14994z do not have complex multiplication.Modular form 14994.2.a.z
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.