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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 54150z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54150.t1 | 54150z1 | \([1, 0, 1, -27435286, -55318180312]\) | \(-14899652746105/1492992\) | \(-228840821859170764800\) | \([]\) | \(4333824\) | \(2.9408\) | \(\Gamma_0(N)\)-optimal |
54150.t2 | 54150z2 | \([1, 0, 1, 1886939, -165616661872]\) | \(4847542295/77309411328\) | \(-11849728080088979944243200\) | \([]\) | \(13001472\) | \(3.4901\) |
Rank
sage: E.rank()
The elliptic curves in class 54150z have rank \(1\).
Complex multiplication
The elliptic curves in class 54150z do not have complex multiplication.Modular form 54150.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.