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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 10830v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10830.q8 | 10830v1 | \([1, 1, 1, 534, -14361]\) | \(357911/2160\) | \(-101619102960\) | \([2]\) | \(13824\) | \(0.79393\) | \(\Gamma_0(N)\)-optimal |
10830.q6 | 10830v2 | \([1, 1, 1, -6686, -193417]\) | \(702595369/72900\) | \(3429644724900\) | \([2, 2]\) | \(27648\) | \(1.1405\) | |
10830.q7 | 10830v3 | \([1, 1, 1, -4881, 427503]\) | \(-273359449/1536000\) | \(-72262473216000\) | \([2]\) | \(41472\) | \(1.3432\) | |
10830.q4 | 10830v4 | \([1, 1, 1, -104156, -12981481]\) | \(2656166199049/33750\) | \(1587798483750\) | \([2]\) | \(55296\) | \(1.4871\) | |
10830.q5 | 10830v5 | \([1, 1, 1, -24736, 1279463]\) | \(35578826569/5314410\) | \(250021100445210\) | \([2]\) | \(55296\) | \(1.4871\) | |
10830.q3 | 10830v6 | \([1, 1, 1, -120401, 15999599]\) | \(4102915888729/9000000\) | \(423412929000000\) | \([2, 2]\) | \(82944\) | \(1.6898\) | |
10830.q2 | 10830v7 | \([1, 1, 1, -163721, 3402143]\) | \(10316097499609/5859375000\) | \(275659458984375000\) | \([2]\) | \(165888\) | \(2.0364\) | |
10830.q1 | 10830v8 | \([1, 1, 1, -1925401, 1027521599]\) | \(16778985534208729/81000\) | \(3810716361000\) | \([2]\) | \(165888\) | \(2.0364\) |
Rank
sage: E.rank()
The elliptic curves in class 10830v have rank \(1\).
Complex multiplication
The elliptic curves in class 10830v do not have complex multiplication.Modular form 10830.2.a.v
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.