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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 6720.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.m1 | 6720bm5 | \([0, -1, 0, -41921, 3317601]\) | \(62161150998242/1607445\) | \(210691031040\) | \([2]\) | \(16384\) | \(1.2795\) | |
6720.m2 | 6720bm3 | \([0, -1, 0, -2721, 48321]\) | \(34008619684/4862025\) | \(318637670400\) | \([2, 2]\) | \(8192\) | \(0.93291\) | |
6720.m3 | 6720bm2 | \([0, -1, 0, -721, -6479]\) | \(2533446736/275625\) | \(4515840000\) | \([2, 2]\) | \(4096\) | \(0.58634\) | |
6720.m4 | 6720bm1 | \([0, -1, 0, -701, -6915]\) | \(37256083456/525\) | \(537600\) | \([2]\) | \(2048\) | \(0.23977\) | \(\Gamma_0(N)\)-optimal |
6720.m5 | 6720bm4 | \([0, -1, 0, 959, -33695]\) | \(1486779836/8203125\) | \(-537600000000\) | \([2]\) | \(8192\) | \(0.93291\) | |
6720.m6 | 6720bm6 | \([0, -1, 0, 4479, 254241]\) | \(75798394558/259416045\) | \(-34002179850240\) | \([2]\) | \(16384\) | \(1.2795\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.m have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.m do not have complex multiplication.Modular form 6720.2.a.m
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.