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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 11830p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11830.u2 | 11830p1 | \([1, 0, 0, -97757631, 403390673945]\) | \(-21405018343206000779641/2177246093750000000\) | \(-10509151040527343750000000\) | \([]\) | \(3556224\) | \(3.5410\) | \(\Gamma_0(N)\)-optimal |
11830.u3 | 11830p2 | \([1, 0, 0, 602007994, -369966529180]\) | \(4998853083179567995470359/2905108466204672000000\) | \(-14022403690652906651648000000\) | \([]\) | \(10668672\) | \(4.0903\) | |
11830.u1 | 11830p3 | \([1, 0, 0, -8535040381, -321869146096655]\) | \(-14245586655234650511684983641/1028175397808386133196800\) | \(-4962806263720098463189513011200\) | \([]\) | \(32006016\) | \(4.6396\) |
Rank
sage: E.rank()
The elliptic curves in class 11830p have rank \(0\).
Complex multiplication
The elliptic curves in class 11830p do not have complex multiplication.Modular form 11830.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.