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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 381150.gp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
381150.gp1 | 381150gp8 | \([1, -1, 0, -144945927792, -21240092336707634]\) | \(16689299266861680229173649/2396798250\) | \(48365440125472722656250\) | \([2]\) | \(955514880\) | \(4.6754\) | |
381150.gp2 | 381150gp7 | \([1, -1, 0, -9297365292, -313497736395134]\) | \(4404531606962679693649/444872222400201750\) | \(8977159773870388425980402343750\) | \([2]\) | \(955514880\) | \(4.6754\) | |
381150.gp3 | 381150gp6 | \([1, -1, 0, -9059146542, -331872739676384]\) | \(4074571110566294433649/48828650062500\) | \(985322461331696844726562500\) | \([2, 2]\) | \(477757440\) | \(4.3288\) | |
381150.gp4 | 381150gp4 | \([1, -1, 0, -2042175042, 35451444281116]\) | \(46676570542430835889/106752955783320\) | \(2154187859222360694954375000\) | \([2]\) | \(318504960\) | \(4.1261\) | |
381150.gp5 | 381150gp5 | \([1, -1, 0, -1791705042, -29058617928884]\) | \(31522423139920199089/164434491947880\) | \(3318154364835553710245625000\) | \([2]\) | \(318504960\) | \(4.1261\) | |
381150.gp6 | 381150gp3 | \([1, -1, 0, -551334042, -5470513113884]\) | \(-918468938249433649/109183593750000\) | \(-2203236157319274902343750000\) | \([2]\) | \(238878720\) | \(3.9822\) | |
381150.gp7 | 381150gp2 | \([1, -1, 0, -174540042, 110187176116]\) | \(29141055407581489/16604321025600\) | \(335061699243056703225000000\) | \([2, 2]\) | \(159252480\) | \(3.7795\) | |
381150.gp8 | 381150gp1 | \([1, -1, 0, 43259958, 13701776116]\) | \(443688652450511/260789760000\) | \(-5262525338799960000000000\) | \([2]\) | \(79626240\) | \(3.4329\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 381150.gp have rank \(0\).
Complex multiplication
The elliptic curves in class 381150.gp do not have complex multiplication.Modular form 381150.2.a.gp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 4 & 6 & 12 \\ 4 & 1 & 2 & 3 & 12 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 12 & 3 & 6 & 1 & 4 & 12 & 2 & 4 \\ 3 & 12 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.