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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 15870u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
15870.u8 | 15870u1 | \([1, 0, 1, 782, -25804]\) | \(357911/2160\) | \(-319757520240\) | \([2]\) | \(25344\) | \(0.88946\) | \(\Gamma_0(N)\)-optimal |
15870.u6 | 15870u2 | \([1, 0, 1, -9798, -338972]\) | \(702595369/72900\) | \(10791816308100\) | \([2, 2]\) | \(50688\) | \(1.2360\) | |
15870.u7 | 15870u3 | \([1, 0, 1, -7153, 761348]\) | \(-273359449/1536000\) | \(-227383125504000\) | \([2]\) | \(76032\) | \(1.4388\) | |
15870.u4 | 15870u4 | \([1, 0, 1, -152628, -22963244]\) | \(2656166199049/33750\) | \(4996211253750\) | \([2]\) | \(101376\) | \(1.5826\) | |
15870.u5 | 15870u5 | \([1, 0, 1, -36248, 2284868]\) | \(35578826569/5314410\) | \(786723408860490\) | \([2]\) | \(101376\) | \(1.5826\) | |
15870.u3 | 15870u6 | \([1, 0, 1, -176433, 28455556]\) | \(4102915888729/9000000\) | \(1332323001000000\) | \([2, 2]\) | \(152064\) | \(1.7853\) | |
15870.u2 | 15870u7 | \([1, 0, 1, -239913, 6135988]\) | \(10316097499609/5859375000\) | \(867397787109375000\) | \([2]\) | \(304128\) | \(2.1319\) | |
15870.u1 | 15870u8 | \([1, 0, 1, -2821433, 1823881556]\) | \(16778985534208729/81000\) | \(11990907009000\) | \([2]\) | \(304128\) | \(2.1319\) |
Rank
sage: E.rank()
The elliptic curves in class 15870u have rank \(0\).
Complex multiplication
The elliptic curves in class 15870u do not have complex multiplication.Modular form 15870.2.a.u
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.