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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 177450ie
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177450.n2 | 177450ie1 | \([1, 1, 0, -8660576200, 309711174664000]\) | \(7620332490460668835709/14321718152921088\) | \(135016011867154070016000000000\) | \([2]\) | \(278691840\) | \(4.4798\) | \(\Gamma_0(N)\)-optimal |
177450.n1 | 177450ie2 | \([1, 1, 0, -138506656200, 19840509297864000]\) | \(31170623789533264459847549/110408848962048\) | \(1040864112987605361000000000\) | \([2]\) | \(557383680\) | \(4.8264\) |
Rank
sage: E.rank()
The elliptic curves in class 177450ie have rank \(0\).
Complex multiplication
The elliptic curves in class 177450ie do not have complex multiplication.Modular form 177450.2.a.ie
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.