Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 122010i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122010.k4 | 122010i1 | \([1, 1, 0, 5176433, 1438819621]\) | \(130384850244802923671/83033078421600000\) | \(-9768758643222818400000\) | \([2]\) | \(10321920\) | \(2.9084\) | \(\Gamma_0(N)\)-optimal |
122010.k3 | 122010i2 | \([1, 1, 0, -21828447, 11770886709]\) | \(9776964066308570159209/5148417725156250000\) | \(605706196946907656250000\) | \([2, 2]\) | \(20643840\) | \(3.2550\) | |
122010.k2 | 122010i3 | \([1, 1, 0, -199719027, -1077737759559]\) | \(7488482171405468850635689/69244766235351562500\) | \(8146577502822875976562500\) | \([2]\) | \(41287680\) | \(3.6016\) | |
122010.k1 | 122010i4 | \([1, 1, 0, -276015947, 1763071924209]\) | \(19766874175324764437159209/23734672172022037500\) | \(2792360446366220689837500\) | \([2]\) | \(41287680\) | \(3.6016\) |
Rank
sage: E.rank()
The elliptic curves in class 122010i have rank \(0\).
Complex multiplication
The elliptic curves in class 122010i do not have complex multiplication.Modular form 122010.2.a.i
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.