Show commands:
SageMath
E = EllipticCurve("ik1")
E.isogeny_class()
Elliptic curves in class 331200ik
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331200.ik4 | 331200ik1 | \([0, 0, 0, -3560700, 16553194000]\) | \(-26752376766544/618796614375\) | \(-115482299361120000000000\) | \([2]\) | \(18874368\) | \(3.1060\) | \(\Gamma_0(N)\)-optimal |
331200.ik3 | 331200ik2 | \([0, 0, 0, -121658700, 514218166000]\) | \(266763091319403556/1355769140625\) | \(1012076240400000000000000\) | \([2, 2]\) | \(37748736\) | \(3.4526\) | |
331200.ik1 | 331200ik3 | \([0, 0, 0, -1944158700, 32994813166000]\) | \(544328872410114151778/14166950625\) | \(21151143947520000000000\) | \([2]\) | \(75497472\) | \(3.7992\) | |
331200.ik2 | 331200ik4 | \([0, 0, 0, -188726700, -115818626000]\) | \(497927680189263938/284271240234375\) | \(424414687500000000000000000\) | \([2]\) | \(75497472\) | \(3.7992\) |
Rank
sage: E.rank()
The elliptic curves in class 331200ik have rank \(0\).
Complex multiplication
The elliptic curves in class 331200ik do not have complex multiplication.Modular form 331200.2.a.ik
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.