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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 336b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
336.c3 | 336b1 | \([0, -1, 0, -7, 10]\) | \(2725888/21\) | \(336\) | \([2]\) | \(16\) | \(-0.68619\) | \(\Gamma_0(N)\)-optimal |
336.c2 | 336b2 | \([0, -1, 0, -12, 0]\) | \(810448/441\) | \(112896\) | \([2, 2]\) | \(32\) | \(-0.33961\) | |
336.c1 | 336b3 | \([0, -1, 0, -152, -672]\) | \(381775972/567\) | \(580608\) | \([2]\) | \(64\) | \(0.0069590\) | |
336.c4 | 336b4 | \([0, -1, 0, 48, -48]\) | \(11696828/7203\) | \(-7375872\) | \([4]\) | \(64\) | \(0.0069590\) |
Rank
sage: E.rank()
The elliptic curves in class 336b have rank \(0\).
Complex multiplication
The elliptic curves in class 336b do not have complex multiplication.Modular form 336.2.a.b
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.