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SageMath
E = EllipticCurve("gz1")
E.isogeny_class()
Elliptic curves in class 424830gz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
424830.gz7 | 424830gz1 | \([1, 0, 0, -457768781, -4493470527855]\) | \(-3735772816268612449/909650165760000\) | \(-2583188932725121021378560000\) | \([2]\) | \(254803968\) | \(3.9783\) | \(\Gamma_0(N)\)-optimal* |
424830.gz6 | 424830gz2 | \([1, 0, 0, -7708200781, -260472872460655]\) | \(17836145204788591940449/770635366502400\) | \(2188420147488642321539174400\) | \([2, 2]\) | \(509607936\) | \(4.3249\) | \(\Gamma_0(N)\)-optimal* |
424830.gz8 | 424830gz3 | \([1, 0, 0, 3294329779, 30569042214801]\) | \(1392333139184610040991/947901937500000000\) | \(-2691814816757517006937500000000\) | \([2]\) | \(764411904\) | \(4.5276\) | \(\Gamma_0(N)\)-optimal* |
424830.gz5 | 424830gz4 | \([1, 0, 0, -8093379981, -233002431167535]\) | \(20645800966247918737249/3688936444974392640\) | \(10475697314057732837509911531840\) | \([2]\) | \(1019215872\) | \(4.6715\) | \(\Gamma_0(N)\)-optimal* |
424830.gz3 | 424830gz5 | \([1, 0, 0, -123329933581, -16670596035724975]\) | \(73054578035931991395831649/136386452160\) | \(387304908034622789856960\) | \([2]\) | \(1019215872\) | \(4.6715\) | |
424830.gz4 | 424830gz6 | \([1, 0, 0, -14406920221, 254939466464801]\) | \(116454264690812369959009/57505157319440250000\) | \(163300894732522101668176460250000\) | \([2, 2]\) | \(1528823808\) | \(4.8742\) | \(\Gamma_0(N)\)-optimal* |
424830.gz1 | 424830gz7 | \([1, 0, 0, -188339402721, 31436754839645301]\) | \(260174968233082037895439009/223081361502731896500\) | \(633497717938247877822879170416500\) | \([2]\) | \(3057647616\) | \(5.2208\) | \(\Gamma_0(N)\)-optimal* |
424830.gz2 | 424830gz8 | \([1, 0, 0, -123694437721, -16567097949715699]\) | \(73704237235978088924479009/899277423164136103500\) | \(2553732824339480902520650737583500\) | \([2]\) | \(3057647616\) | \(5.2208\) |
Rank
sage: E.rank()
The elliptic curves in class 424830gz have rank \(0\).
Complex multiplication
The elliptic curves in class 424830gz do not have complex multiplication.Modular form 424830.2.a.gz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.