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SageMath
E = EllipticCurve("rj1")
E.isogeny_class()
Elliptic curves in class 176400.rj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
176400.rj1 | 176400gp5 | \([0, 0, 0, -2963523675, 62095583898250]\) | \(524388516989299201/3150\) | \(17290449993600000000\) | \([2]\) | \(56623104\) | \(3.7569\) | |
176400.rj2 | 176400gp3 | \([0, 0, 0, -185223675, 970205598250]\) | \(128031684631201/9922500\) | \(54464917479840000000000\) | \([2, 2]\) | \(28311552\) | \(3.4103\) | |
176400.rj3 | 176400gp6 | \([0, 0, 0, -172875675, 1105132194250]\) | \(-104094944089921/35880468750\) | \(-196949031958350000000000000\) | \([2]\) | \(56623104\) | \(3.7569\) | |
176400.rj4 | 176400gp4 | \([0, 0, 0, -65271675, -191839985750]\) | \(5602762882081/345888060\) | \(1898590541210576640000000\) | \([2]\) | \(28311552\) | \(3.4103\) | |
176400.rj5 | 176400gp2 | \([0, 0, 0, -12351675, 13013334250]\) | \(37966934881/8643600\) | \(47444994782438400000000\) | \([2, 2]\) | \(14155776\) | \(3.0637\) | |
176400.rj6 | 176400gp1 | \([0, 0, 0, 1760325, 1258038250]\) | \(109902239/188160\) | \(-1032816212951040000000\) | \([2]\) | \(7077888\) | \(2.7172\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 176400.rj have rank \(0\).
Complex multiplication
The elliptic curves in class 176400.rj do not have complex multiplication.Modular form 176400.2.a.rj
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.