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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 188370.ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
188370.ca1 | 188370u7 | \([1, -1, 1, -4488970883, 115763459349251]\) | \(13722604572968640187892492722921/36939806611960382108160\) | \(26929119020119118556848640\) | \([6]\) | \(159252480\) | \(4.1139\) | |
188370.ca2 | 188370u8 | \([1, -1, 1, -797732483, -6404131470589]\) | \(77013704252633562960444236521/20262661472595628847255040\) | \(14771480213522213429648924160\) | \([6]\) | \(159252480\) | \(4.1139\) | |
188370.ca3 | 188370u5 | \([1, -1, 1, -738264308, -7720679532769]\) | \(61042428203425827148268361721/2287149206968899000\) | \(1667331771880327371000\) | \([2]\) | \(53084160\) | \(3.5646\) | |
188370.ca4 | 188370u6 | \([1, -1, 1, -284055683, 1761480412931]\) | \(3477015524751011858387583721/173605868128473455001600\) | \(126558677865657148696166400\) | \([2, 6]\) | \(79626240\) | \(3.7674\) | |
188370.ca5 | 188370u4 | \([1, -1, 1, -74154308, 42318131231]\) | \(61859347930211625693801721/34737934177406743101000\) | \(25323954015329515720629000\) | \([2]\) | \(53084160\) | \(3.5646\) | |
188370.ca6 | 188370u2 | \([1, -1, 1, -46209308, -120254700769]\) | \(14968716721822395621081721/91209357028881000000\) | \(66491621274054249000000\) | \([2, 2]\) | \(26542080\) | \(3.2181\) | |
188370.ca7 | 188370u1 | \([1, -1, 1, -1209308, -4046700769]\) | \(-268291321601301081721/9550359000000000000\) | \(-6962211711000000000000\) | \([2]\) | \(13271040\) | \(2.8715\) | \(\Gamma_0(N)\)-optimal |
188370.ca8 | 188370u3 | \([1, -1, 1, 10856317, 107731881731]\) | \(194108149567956675968279/6990401110687088640000\) | \(-5096002409690887618560000\) | \([6]\) | \(39813120\) | \(3.4208\) |
Rank
sage: E.rank()
The elliptic curves in class 188370.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 188370.ca do not have complex multiplication.Modular form 188370.2.a.ca
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.