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SageMath
E = EllipticCurve("ir1")
E.isogeny_class()
Elliptic curves in class 177600ir
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177600.ci5 | 177600ir1 | \([0, -1, 0, -1352033, 2083475937]\) | \(-66730743078481/419010969600\) | \(-1716268931481600000000\) | \([2]\) | \(7962624\) | \(2.7581\) | \(\Gamma_0(N)\)-optimal |
| 177600.ci4 | 177600ir2 | \([0, -1, 0, -34120033, 76565139937]\) | \(1072487167529950801/2554882560000\) | \(10464798965760000000000\) | \([2, 2]\) | \(15925248\) | \(3.1047\) | |
| 177600.ci1 | 177600ir3 | \([0, -1, 0, -545608033, 4905523347937]\) | \(4385367890843575421521/24975000000\) | \(102297600000000000000\) | \([2]\) | \(31850496\) | \(3.4513\) | |
| 177600.ci3 | 177600ir4 | \([0, -1, 0, -46920033, 13909139937]\) | \(2788936974993502801/1593609593601600\) | \(6527424895392153600000000\) | \([2, 2]\) | \(31850496\) | \(3.4513\) | |
| 177600.ci6 | 177600ir5 | \([0, -1, 0, 186359967, 110720339937]\) | \(174751791402194852399/102423900876336360\) | \(-419528297989473730560000000\) | \([4]\) | \(63700992\) | \(3.7978\) | |
| 177600.ci2 | 177600ir6 | \([0, -1, 0, -485000033, -4093090860063]\) | \(3080272010107543650001/15465841417699560\) | \(63348086446897397760000000\) | \([2]\) | \(63700992\) | \(3.7978\) |
Rank
sage: E.rank()
The elliptic curves in class 177600ir have rank \(0\).
Complex multiplication
The elliptic curves in class 177600ir do not have complex multiplication.Modular form 177600.2.a.ir
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
