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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 11011p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11011.m4 | 11011p1 | \([1, -1, 0, -1898, 268879]\) | \(-426957777/17320303\) | \(-30683973302983\) | \([2]\) | \(18240\) | \(1.2680\) | \(\Gamma_0(N)\)-optimal |
11011.m3 | 11011p2 | \([1, -1, 0, -75103, 7896840]\) | \(26444947540257/169338169\) | \(299992896011809\) | \([2, 2]\) | \(36480\) | \(1.6146\) | |
11011.m2 | 11011p3 | \([1, -1, 0, -121688, -3032001]\) | \(112489728522417/62811265517\) | \(111273988350562037\) | \([2]\) | \(72960\) | \(1.9612\) | |
11011.m1 | 11011p4 | \([1, -1, 0, -1199798, 506136725]\) | \(107818231938348177/4463459\) | \(7907289889499\) | \([4]\) | \(72960\) | \(1.9612\) |
Rank
sage: E.rank()
The elliptic curves in class 11011p have rank \(0\).
Complex multiplication
The elliptic curves in class 11011p do not have complex multiplication.Modular form 11011.2.a.p
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.