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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 13454.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13454.d1 | 13454d6 | \([1, 1, 0, -2624030, 1634974964]\) | \(2251439055699625/25088\) | \(22265692348928\) | \([2]\) | \(181440\) | \(2.1301\) | |
13454.d2 | 13454d5 | \([1, 1, 0, -163870, 25538292]\) | \(-548347731625/1835008\) | \(-1628576354664448\) | \([2]\) | \(90720\) | \(1.7835\) | |
13454.d3 | 13454d4 | \([1, 1, 0, -34135, 1975533]\) | \(4956477625/941192\) | \(835311364527752\) | \([2]\) | \(60480\) | \(1.5808\) | |
13454.d4 | 13454d2 | \([1, 1, 0, -10110, -395254]\) | \(128787625/98\) | \(86975360738\) | \([2]\) | \(20160\) | \(1.0315\) | |
13454.d5 | 13454d1 | \([1, 1, 0, -500, -8932]\) | \(-15625/28\) | \(-24850103068\) | \([2]\) | \(10080\) | \(0.68491\) | \(\Gamma_0(N)\)-optimal |
13454.d6 | 13454d3 | \([1, 1, 0, 4305, 184229]\) | \(9938375/21952\) | \(-19482480805312\) | \([2]\) | \(30240\) | \(1.2342\) |
Rank
sage: E.rank()
The elliptic curves in class 13454.d have rank \(1\).
Complex multiplication
The elliptic curves in class 13454.d do not have complex multiplication.Modular form 13454.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.