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SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 431970.bq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
431970.bq1 | 431970bq7 | \([1, 0, 1, -1609283789, 24829773089636]\) | \(260174968233082037895439009/223081361502731896500\) | \(395202239865141221295436500\) | \([2]\) | \(318504960\) | \(4.0302\) | \(\Gamma_0(N)\)-optimal* |
431970.bq2 | 431970bq8 | \([1, 0, 1, -1056918789, -13085377744364]\) | \(73704237235978088924479009/899277423164136103500\) | \(1593124811058080119652563500\) | \([2]\) | \(318504960\) | \(4.0302\) | |
431970.bq3 | 431970bq5 | \([1, 0, 1, -1053804249, -13167124069508]\) | \(73054578035931991395831649/136386452160\) | \(241616919575021760\) | \([2]\) | \(106168320\) | \(3.4809\) | |
431970.bq4 | 431970bq6 | \([1, 0, 1, -123101289, 201351172636]\) | \(116454264690812369959009/57505157319440250000\) | \(101873894005984888730250000\) | \([2, 2]\) | \(159252480\) | \(3.6836\) | \(\Gamma_0(N)\)-optimal* |
431970.bq5 | 431970bq4 | \([1, 0, 1, -69154649, -184039042948]\) | \(20645800966247918737249/3688936444974392640\) | \(6535175937395279999711040\) | \([2]\) | \(106168320\) | \(3.4809\) | \(\Gamma_0(N)\)-optimal* |
431970.bq6 | 431970bq2 | \([1, 0, 1, -65863449, -205735949828]\) | \(17836145204788591940449/770635366502400\) | \(1365227560516358246400\) | \([2, 2]\) | \(53084160\) | \(3.1343\) | \(\Gamma_0(N)\)-optimal* |
431970.bq7 | 431970bq1 | \([1, 0, 1, -3911449, -3549402628]\) | \(-3735772816268612449/909650165760000\) | \(-1611500757303951360000\) | \([2]\) | \(26542080\) | \(2.7877\) | \(\Gamma_0(N)\)-optimal* |
431970.bq8 | 431970bq3 | \([1, 0, 1, 28148711, 24146672636]\) | \(1392333139184610040991/947901937500000000\) | \(-1679266104299437500000000\) | \([2]\) | \(79626240\) | \(3.3370\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 431970.bq have rank \(0\).
Complex multiplication
The elliptic curves in class 431970.bq do not have complex multiplication.Modular form 431970.2.a.bq
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.