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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 39270.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39270.cp1 | 39270cn8 | \([1, 0, 0, -1333355333356, -592607370447828664]\) | \(262156976355489363181342849900999019467969/296485141924125000\) | \(296485141924125000\) | \([2]\) | \(222953472\) | \(5.0236\) | |
| 39270.cp2 | 39270cn6 | \([1, 0, 0, -83334708356, -9259495366703664]\) | \(64003168104546012500462338813649467969/68064746081030015625000000\) | \(68064746081030015625000000\) | \([2, 2]\) | \(111476736\) | \(4.6770\) | |
| 39270.cp3 | 39270cn7 | \([1, 0, 0, -83314083356, -9264307785578664]\) | \(-63955658296770964115513956628279467969/66004356107812185925891924125000\) | \(-66004356107812185925891924125000\) | \([2]\) | \(222953472\) | \(5.0236\) | |
| 39270.cp4 | 39270cn5 | \([1, 0, 0, -16461218116, -812901436975600]\) | \(493298302018650738343048153196947009/5139490792463830279120089600\) | \(5139490792463830279120089600\) | \([6]\) | \(74317824\) | \(4.4743\) | |
| 39270.cp5 | 39270cn3 | \([1, 0, 0, -5209708356, -144604741703664]\) | \(15637378471582822120727563649467969/16113547119140625000000000000\) | \(16113547119140625000000000000\) | \([2]\) | \(55738368\) | \(4.3305\) | |
| 39270.cp6 | 39270cn2 | \([1, 0, 0, -1054050116, -12046088636400]\) | \(129511249478743944259581330835009/12262789317997149185802240000\) | \(12262789317997149185802240000\) | \([2, 6]\) | \(37158912\) | \(4.1277\) | |
| 39270.cp7 | 39270cn1 | \([1, 0, 0, -234850116, 1175307843600]\) | \(1432504679512464302827718035009/232233326153721446400000000\) | \(232233326153721446400000000\) | \([6]\) | \(18579456\) | \(3.7812\) | \(\Gamma_0(N)\)-optimal |
| 39270.cp8 | 39270cn4 | \([1, 0, 0, 1245917884, -57356838217200]\) | \(213890734289719241265598586476991/1544981081981970035652027609600\) | \(-1544981081981970035652027609600\) | \([6]\) | \(74317824\) | \(4.4743\) |
Rank
sage: E.rank()
The elliptic curves in class 39270.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 39270.cp do not have complex multiplication.Modular form 39270.2.a.cp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
