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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 134310.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134310.r1 | 134310ch4 | \([1, 1, 0, -41261607, -102032965899]\) | \(4385367890843575421521/24975000000\) | \(44244735975000000\) | \([2]\) | \(8847360\) | \(2.8058\) | |
| 134310.r2 | 134310ch5 | \([1, 1, 0, -36678127, 85111578589]\) | \(3080272010107543650001/15465841417699560\) | \(27398681487781250213160\) | \([2]\) | \(17694720\) | \(3.1523\) | |
| 134310.r3 | 134310ch3 | \([1, 1, 0, -3548327, -290419851]\) | \(2788936974993502801/1593609593601600\) | \(2823176605250444097600\) | \([2, 2]\) | \(8847360\) | \(2.8058\) | |
| 134310.r4 | 134310ch2 | \([1, 1, 0, -2580327, -1593154251]\) | \(1072487167529950801/2554882560000\) | \(4526130302876160000\) | \([2, 2]\) | \(4423680\) | \(2.4592\) | |
| 134310.r5 | 134310ch1 | \([1, 1, 0, -102247, -43363019]\) | \(-66730743078481/419010969600\) | \(-742303492315545600\) | \([2]\) | \(2211840\) | \(2.1126\) | \(\Gamma_0(N)\)-optimal |
| 134310.r6 | 134310ch6 | \([1, 1, 0, 14093473, -2298056691]\) | \(174751791402194852399/102423900876336360\) | \(-181450188260383318257960\) | \([2]\) | \(17694720\) | \(3.1523\) |
Rank
sage: E.rank()
The elliptic curves in class 134310.r have rank \(0\).
Complex multiplication
The elliptic curves in class 134310.r do not have complex multiplication.Modular form 134310.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
