Show commands:
SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 57222.bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
57222.bi1 | 57222bu4 | \([1, -1, 1, -4125983261, -102008160155635]\) | \(441453577446719855661097/4354701912\) | \(76626588131116975512\) | \([2]\) | \(24772608\) | \(3.8472\) | |
57222.bi2 | 57222bu2 | \([1, -1, 1, -257880101, -1593749363299]\) | \(107784459654566688937/10704361149504\) | \(188357019512515572400704\) | \([2, 2]\) | \(12386304\) | \(3.5007\) | |
57222.bi3 | 57222bu3 | \([1, -1, 1, -238424621, -1844374856659]\) | \(-85183593440646799657/34223681512621656\) | \(-602209749505853673001218456\) | \([2]\) | \(24772608\) | \(3.8472\) | |
57222.bi4 | 57222bu1 | \([1, -1, 1, -17339621, -20903272675]\) | \(32765849647039657/8229948198912\) | \(144816537095377147072512\) | \([2]\) | \(6193152\) | \(3.1541\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 57222.bi have rank \(0\).
Complex multiplication
The elliptic curves in class 57222.bi do not have complex multiplication.Modular form 57222.2.a.bi
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.