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SageMath
E = EllipticCurve("eo1")
E.isogeny_class()
Elliptic curves in class 33600.eo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.eo1 | 33600cc8 | \([0, 1, 0, -10321633, 8026548863]\) | \(29689921233686449/10380965400750\) | \(42520434281472000000000\) | \([2]\) | \(2654208\) | \(3.0435\) | |
33600.eo2 | 33600cc5 | \([0, 1, 0, -9217633, 10768452863]\) | \(21145699168383889/2593080\) | \(10621255680000000\) | \([2]\) | \(884736\) | \(2.4942\) | |
33600.eo3 | 33600cc6 | \([0, 1, 0, -4321633, -3367451137]\) | \(2179252305146449/66177562500\) | \(271063296000000000000\) | \([2, 2]\) | \(1327104\) | \(2.6969\) | |
33600.eo4 | 33600cc3 | \([0, 1, 0, -4289633, -3421051137]\) | \(2131200347946769/2058000\) | \(8429568000000000\) | \([2]\) | \(663552\) | \(2.3503\) | |
33600.eo5 | 33600cc2 | \([0, 1, 0, -577633, 167172863]\) | \(5203798902289/57153600\) | \(234101145600000000\) | \([2, 2]\) | \(442368\) | \(2.1476\) | |
33600.eo6 | 33600cc4 | \([0, 1, 0, -129633, 420292863]\) | \(-58818484369/18600435000\) | \(-76187381760000000000\) | \([2]\) | \(884736\) | \(2.4942\) | |
33600.eo7 | 33600cc1 | \([0, 1, 0, -65633, -2299137]\) | \(7633736209/3870720\) | \(15854469120000000\) | \([2]\) | \(221184\) | \(1.8010\) | \(\Gamma_0(N)\)-optimal |
33600.eo8 | 33600cc7 | \([0, 1, 0, 1166367, -11330539137]\) | \(42841933504271/13565917968750\) | \(-55566000000000000000000\) | \([2]\) | \(2654208\) | \(3.0435\) |
Rank
sage: E.rank()
The elliptic curves in class 33600.eo have rank \(0\).
Complex multiplication
The elliptic curves in class 33600.eo do not have complex multiplication.Modular form 33600.2.a.eo
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.