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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 1470.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1470.q1 | 1470p5 | \([1, 0, 0, -823201, 287410955]\) | \(524388516989299201/3150\) | \(370594350\) | \([2]\) | \(12288\) | \(1.7097\) | |
1470.q2 | 1470p4 | \([1, 0, 0, -51451, 4487405]\) | \(128031684631201/9922500\) | \(1167372202500\) | \([2, 2]\) | \(6144\) | \(1.3631\) | |
1470.q3 | 1470p6 | \([1, 0, 0, -48021, 5112351]\) | \(-104094944089921/35880468750\) | \(-4221301267968750\) | \([2]\) | \(12288\) | \(1.7097\) | |
1470.q4 | 1470p3 | \([1, 0, 0, -18131, -889659]\) | \(5602762882081/345888060\) | \(40693384370940\) | \([2]\) | \(6144\) | \(1.3631\) | |
1470.q5 | 1470p2 | \([1, 0, 0, -3431, 59961]\) | \(37966934881/8643600\) | \(1016910896400\) | \([2, 2]\) | \(3072\) | \(1.0166\) | |
1470.q6 | 1470p1 | \([1, 0, 0, 489, 5865]\) | \(109902239/188160\) | \(-22136835840\) | \([2]\) | \(1536\) | \(0.66999\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1470.q have rank \(0\).
Complex multiplication
The elliptic curves in class 1470.q do not have complex multiplication.Modular form 1470.2.a.q
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.