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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1008.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1008.j1 | 1008l4 | \([0, 0, 0, -193539, 32771842]\) | \(268498407453697/252\) | \(752467968\) | \([2]\) | \(3072\) | \(1.4316\) | |
1008.j2 | 1008l5 | \([0, 0, 0, -131619, -18202718]\) | \(84448510979617/933897762\) | \(2788603774967808\) | \([2]\) | \(6144\) | \(1.7781\) | |
1008.j3 | 1008l3 | \([0, 0, 0, -14979, 249730]\) | \(124475734657/63011844\) | \(188152357994496\) | \([2, 2]\) | \(3072\) | \(1.4316\) | |
1008.j4 | 1008l2 | \([0, 0, 0, -12099, 511810]\) | \(65597103937/63504\) | \(189621927936\) | \([2, 2]\) | \(1536\) | \(1.0850\) | |
1008.j5 | 1008l1 | \([0, 0, 0, -579, 11842]\) | \(-7189057/16128\) | \(-48157949952\) | \([2]\) | \(768\) | \(0.73841\) | \(\Gamma_0(N)\)-optimal |
1008.j6 | 1008l6 | \([0, 0, 0, 55581, 1929058]\) | \(6359387729183/4218578658\) | \(-12596608375529472\) | \([2]\) | \(6144\) | \(1.7781\) |
Rank
sage: E.rank()
The elliptic curves in class 1008.j have rank \(0\).
Complex multiplication
The elliptic curves in class 1008.j do not have complex multiplication.Modular form 1008.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.