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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 42042.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42042.q1 | 42042t4 | \([1, 1, 0, -298447559, -1984618689675]\) | \(24988464356366680777409257/19820013127858944\) | \(2331804724479476902656\) | \([2]\) | \(11796480\) | \(3.4062\) | |
42042.q2 | 42042t2 | \([1, 1, 0, -18779079, -30575019915]\) | \(6225272619854317474537/171699142176866304\) | \(20200232377966143799296\) | \([2, 2]\) | \(5898240\) | \(3.0596\) | |
42042.q3 | 42042t1 | \([1, 1, 0, -2722759, 1046296693]\) | \(18974193623767438057/6951907079749632\) | \(817884916025464455168\) | \([2]\) | \(2949120\) | \(2.7131\) | \(\Gamma_0(N)\)-optimal |
42042.q4 | 42042t3 | \([1, 1, 0, 3988281, -99974486667]\) | \(59633809076653006103/36736390236271934208\) | \(-4321999574907156787636992\) | \([2]\) | \(11796480\) | \(3.4062\) |
Rank
sage: E.rank()
The elliptic curves in class 42042.q have rank \(1\).
Complex multiplication
The elliptic curves in class 42042.q do not have complex multiplication.Modular form 42042.2.a.q
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.