Show commands:
SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 12342.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12342.w1 | 12342v3 | \([1, 1, 1, -58556682, -172490345301]\) | \(12534210458299016895673/315581882565708\) | \(559072555459988230188\) | \([2]\) | \(1843200\) | \(3.0892\) | |
12342.w2 | 12342v2 | \([1, 1, 1, -3799342, -2479756069]\) | \(3423676911662954233/483711578981136\) | \(856924568571400273296\) | \([2, 2]\) | \(921600\) | \(2.7426\) | |
12342.w3 | 12342v1 | \([1, 1, 1, -1001822, 346858139]\) | \(62768149033310713/6915442583808\) | \(12251128379213484288\) | \([4]\) | \(460800\) | \(2.3961\) | \(\Gamma_0(N)\)-optimal |
12342.w4 | 12342v4 | \([1, 1, 1, 6197678, -13316525749]\) | \(14861225463775641287/51859390496937804\) | \(-91872073688145632992044\) | \([2]\) | \(1843200\) | \(3.0892\) |
Rank
sage: E.rank()
The elliptic curves in class 12342.w have rank \(0\).
Complex multiplication
The elliptic curves in class 12342.w do not have complex multiplication.Modular form 12342.2.a.w
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.