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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 5040.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.i1 | 5040bg5 | \([0, 0, 0, -2419203, -1448293502]\) | \(524388516989299201/3150\) | \(9405849600\) | \([2]\) | \(49152\) | \(1.9792\) | |
5040.i2 | 5040bg3 | \([0, 0, 0, -151203, -22628702]\) | \(128031684631201/9922500\) | \(29628426240000\) | \([2, 2]\) | \(24576\) | \(1.6326\) | |
5040.i3 | 5040bg6 | \([0, 0, 0, -141123, -25775678]\) | \(-104094944089921/35880468750\) | \(-107138505600000000\) | \([2]\) | \(49152\) | \(1.9792\) | |
5040.i4 | 5040bg4 | \([0, 0, 0, -53283, 4474402]\) | \(5602762882081/345888060\) | \(1032816212951040\) | \([2]\) | \(24576\) | \(1.6326\) | |
5040.i5 | 5040bg2 | \([0, 0, 0, -10083, -303518]\) | \(37966934881/8643600\) | \(25809651302400\) | \([2, 2]\) | \(12288\) | \(1.2861\) | |
5040.i6 | 5040bg1 | \([0, 0, 0, 1437, -29342]\) | \(109902239/188160\) | \(-561842749440\) | \([2]\) | \(6144\) | \(0.93949\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5040.i have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.i do not have complex multiplication.Modular form 5040.2.a.i
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.