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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 2730.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2730.o1 | 2730n7 | \([1, 0, 1, -2265679, -1312610494]\) | \(1286229821345376481036009/247265484375000000\) | \(247265484375000000\) | \([2]\) | \(82944\) | \(2.3379\) | |
2730.o2 | 2730n8 | \([1, 0, 1, -996559, 370812482]\) | \(109454124781830273937129/3914078300576808000\) | \(3914078300576808000\) | \([2]\) | \(82944\) | \(2.3379\) | |
2730.o3 | 2730n5 | \([1, 0, 1, -987844, 377820686]\) | \(106607603143751752938169/5290068420\) | \(5290068420\) | \([6]\) | \(27648\) | \(1.7886\) | |
2730.o4 | 2730n6 | \([1, 0, 1, -156559, -15923518]\) | \(424378956393532177129/136231857216000000\) | \(136231857216000000\) | \([2, 2]\) | \(41472\) | \(1.9914\) | |
2730.o5 | 2730n4 | \([1, 0, 1, -68764, 4472462]\) | \(35958207000163259449/12145729518877500\) | \(12145729518877500\) | \([6]\) | \(27648\) | \(1.7886\) | |
2730.o6 | 2730n2 | \([1, 0, 1, -61744, 5898926]\) | \(26031421522845051769/5797789779600\) | \(5797789779600\) | \([2, 6]\) | \(13824\) | \(1.4420\) | |
2730.o7 | 2730n1 | \([1, 0, 1, -3424, 113582]\) | \(-4437543642183289/3033210136320\) | \(-3033210136320\) | \([6]\) | \(6912\) | \(1.0955\) | \(\Gamma_0(N)\)-optimal |
2730.o8 | 2730n3 | \([1, 0, 1, 27761, -1694014]\) | \(2366200373628880151/2612420149248000\) | \(-2612420149248000\) | \([2]\) | \(20736\) | \(1.6448\) |
Rank
sage: E.rank()
The elliptic curves in class 2730.o have rank \(0\).
Complex multiplication
The elliptic curves in class 2730.o do not have complex multiplication.Modular form 2730.2.a.o
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.