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SageMath
E = EllipticCurve("lm1")
E.isogeny_class()
Elliptic curves in class 176400.lm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
176400.lm1 | 176400fm8 | \([0, 0, 0, -1137960075, -9295716487750]\) | \(29689921233686449/10380965400750\) | \(56981448618082431408000000000\) | \([2]\) | \(127401984\) | \(4.2192\) | |
176400.lm2 | 176400fm5 | \([0, 0, 0, -1016244075, -12469387099750]\) | \(21145699168383889/2593080\) | \(14233498434731520000000\) | \([2]\) | \(42467328\) | \(3.6698\) | |
176400.lm3 | 176400fm6 | \([0, 0, 0, -476460075, 3896578012250]\) | \(2179252305146449/66177562500\) | \(363250741303044000000000000\) | \([2, 2]\) | \(63700992\) | \(3.8726\) | |
176400.lm4 | 176400fm3 | \([0, 0, 0, -472932075, 3958639060250]\) | \(2131200347946769/2058000\) | \(11296427329152000000000\) | \([2]\) | \(31850496\) | \(3.5260\) | |
176400.lm5 | 176400fm2 | \([0, 0, 0, -63684075, -193746379750]\) | \(5203798902289/57153600\) | \(313717924683878400000000\) | \([2, 2]\) | \(21233664\) | \(3.3233\) | |
176400.lm6 | 176400fm4 | \([0, 0, 0, -14292075, -486591547750]\) | \(-58818484369/18600435000\) | \(-102098378167208640000000000\) | \([2]\) | \(42467328\) | \(3.6698\) | |
176400.lm7 | 176400fm1 | \([0, 0, 0, -7236075, 2636212250]\) | \(7633736209/3870720\) | \(21246504952135680000000\) | \([2]\) | \(10616832\) | \(2.9767\) | \(\Gamma_0(N)\)-optimal |
176400.lm8 | 176400fm7 | \([0, 0, 0, 128591925, 13116965440250]\) | \(42841933504271/13565917968750\) | \(-74463754366968750000000000000\) | \([2]\) | \(127401984\) | \(4.2192\) |
Rank
sage: E.rank()
The elliptic curves in class 176400.lm have rank \(0\).
Complex multiplication
The elliptic curves in class 176400.lm do not have complex multiplication.Modular form 176400.2.a.lm
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.