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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 55545.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55545.t1 | 55545g4 | \([1, 0, 1, -59524, 5584241]\) | \(157551496201/13125\) | \(1942971043125\) | \([2]\) | \(180224\) | \(1.4018\) | |
55545.t2 | 55545g2 | \([1, 0, 1, -3979, 74177]\) | \(47045881/11025\) | \(1632095676225\) | \([2, 2]\) | \(90112\) | \(1.0552\) | |
55545.t3 | 55545g1 | \([1, 0, 1, -1334, -17869]\) | \(1771561/105\) | \(15543768345\) | \([2]\) | \(45056\) | \(0.70864\) | \(\Gamma_0(N)\)-optimal |
55545.t4 | 55545g3 | \([1, 0, 1, 9246, 465637]\) | \(590589719/972405\) | \(-143950838643045\) | \([2]\) | \(180224\) | \(1.4018\) |
Rank
sage: E.rank()
The elliptic curves in class 55545.t have rank \(1\).
Complex multiplication
The elliptic curves in class 55545.t do not have complex multiplication.Modular form 55545.2.a.t
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.