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SageMath
E = EllipticCurve("ix1")
E.isogeny_class()
Elliptic curves in class 176400.ix
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
176400.ix1 | 176400jz4 | \([0, 0, 0, -4663575, -1553532750]\) | \(1210991472/588245\) | \(5448761119545660000000\) | \([2]\) | \(7962624\) | \(2.8643\) | |
176400.ix2 | 176400jz3 | \([0, 0, 0, -3836700, -2890589625]\) | \(10788913152/8575\) | \(4964250291131250000\) | \([2]\) | \(3981312\) | \(2.5177\) | |
176400.ix3 | 176400jz2 | \([0, 0, 0, -2458575, 1483732250]\) | \(129348709488/6125\) | \(77824813500000000\) | \([2]\) | \(2654208\) | \(2.3150\) | |
176400.ix4 | 176400jz1 | \([0, 0, 0, -161700, 20622875]\) | \(588791808/109375\) | \(86858050781250000\) | \([2]\) | \(1327104\) | \(1.9684\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 176400.ix have rank \(1\).
Complex multiplication
The elliptic curves in class 176400.ix do not have complex multiplication.Modular form 176400.2.a.ix
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.