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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 25350.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25350.d1 | 25350i7 | \([1, 1, 0, -22534125, -41182072875]\) | \(16778985534208729/81000\) | \(6108930140625000\) | \([2]\) | \(1327104\) | \(2.6514\) | |
| 25350.d2 | 25350i8 | \([1, 1, 0, -1916125, -139746875]\) | \(10316097499609/5859375000\) | \(441907562255859375000\) | \([2]\) | \(1327104\) | \(2.6514\) | |
| 25350.d3 | 25350i6 | \([1, 1, 0, -1409125, -643197875]\) | \(4102915888729/9000000\) | \(678770015625000000\) | \([2, 2]\) | \(663552\) | \(2.3048\) | |
| 25350.d4 | 25350i5 | \([1, 1, 0, -1219000, 517515250]\) | \(2656166199049/33750\) | \(2545387558593750\) | \([2]\) | \(442368\) | \(2.1021\) | |
| 25350.d5 | 25350i4 | \([1, 1, 0, -289500, -51761250]\) | \(35578826569/5314410\) | \(400806906526406250\) | \([2]\) | \(442368\) | \(2.1021\) | |
| 25350.d6 | 25350i2 | \([1, 1, 0, -78250, 7600000]\) | \(702595369/72900\) | \(5498037126562500\) | \([2, 2]\) | \(221184\) | \(1.7555\) | |
| 25350.d7 | 25350i3 | \([1, 1, 0, -57125, -17221875]\) | \(-273359449/1536000\) | \(-115843416000000000\) | \([2]\) | \(331776\) | \(1.9582\) | |
| 25350.d8 | 25350i1 | \([1, 1, 0, 6250, 586500]\) | \(357911/2160\) | \(-162904803750000\) | \([2]\) | \(110592\) | \(1.4089\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25350.d have rank \(1\).
Complex multiplication
The elliptic curves in class 25350.d do not have complex multiplication.Modular form 25350.2.a.d
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
