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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 210a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
210.d7 | 210a1 | \([1, 0, 0, -41, -39]\) | \(7633736209/3870720\) | \(3870720\) | \([6]\) | \(48\) | \(-0.043427\) | \(\Gamma_0(N)\)-optimal |
210.d5 | 210a2 | \([1, 0, 0, -361, 2585]\) | \(5203798902289/57153600\) | \(57153600\) | \([2, 6]\) | \(96\) | \(0.30315\) | |
210.d4 | 210a3 | \([1, 0, 0, -2681, -53655]\) | \(2131200347946769/2058000\) | \(2058000\) | \([2]\) | \(144\) | \(0.50588\) | |
210.d2 | 210a4 | \([1, 0, 0, -5761, 167825]\) | \(21145699168383889/2593080\) | \(2593080\) | \([6]\) | \(192\) | \(0.64972\) | |
210.d6 | 210a5 | \([1, 0, 0, -81, 6561]\) | \(-58818484369/18600435000\) | \(-18600435000\) | \([6]\) | \(192\) | \(0.64972\) | |
210.d3 | 210a6 | \([1, 0, 0, -2701, -52819]\) | \(2179252305146449/66177562500\) | \(66177562500\) | \([2, 2]\) | \(288\) | \(0.85245\) | |
210.d1 | 210a7 | \([1, 0, 0, -6451, 124931]\) | \(29689921233686449/10380965400750\) | \(10380965400750\) | \([2]\) | \(576\) | \(1.1990\) | |
210.d8 | 210a8 | \([1, 0, 0, 729, -176985]\) | \(42841933504271/13565917968750\) | \(-13565917968750\) | \([2]\) | \(576\) | \(1.1990\) |
Rank
sage: E.rank()
The elliptic curves in class 210a have rank \(0\).
Complex multiplication
The elliptic curves in class 210a do not have complex multiplication.Modular form 210.2.a.a
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.