Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 147030bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
147030.bo3 | 147030bf1 | \([1, 1, 1, -299556, 62978469]\) | \(615882348586441/21715200\) | \(104815122796800\) | \([4]\) | \(2064384\) | \(1.7801\) | \(\Gamma_0(N)\)-optimal |
147030.bo2 | 147030bf2 | \([1, 1, 1, -313076, 56964773]\) | \(703093388853961/115124490000\) | \(555683924452410000\) | \([2, 2]\) | \(4128768\) | \(2.1266\) | |
147030.bo4 | 147030bf3 | \([1, 1, 1, 569104, 320913029]\) | \(4223169036960119/11647532812500\) | \(-56220416207170312500\) | \([2]\) | \(8257536\) | \(2.4732\) | |
147030.bo1 | 147030bf4 | \([1, 1, 1, -1411576, -591589627]\) | \(64443098670429961/6032611833300\) | \(29118265090478939700\) | \([2]\) | \(8257536\) | \(2.4732\) |
Rank
sage: E.rank()
The elliptic curves in class 147030bf have rank \(1\).
Complex multiplication
The elliptic curves in class 147030bf do not have complex multiplication.Modular form 147030.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.