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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 14a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14.a6 | 14a1 | \([1, 0, 1, 4, -6]\) | \(9938375/21952\) | \(-21952\) | \([6]\) | \(1\) | \(-0.48278\) | \(\Gamma_0(N)\)-optimal |
14.a3 | 14a2 | \([1, 0, 1, -36, -70]\) | \(4956477625/941192\) | \(941192\) | \([6]\) | \(2\) | \(-0.13621\) | |
14.a2 | 14a3 | \([1, 0, 1, -171, -874]\) | \(-548347731625/1835008\) | \(-1835008\) | \([2]\) | \(3\) | \(0.066527\) | |
14.a5 | 14a4 | \([1, 0, 1, -1, 0]\) | \(-15625/28\) | \(-28\) | \([6]\) | \(3\) | \(-1.0321\) | |
14.a1 | 14a5 | \([1, 0, 1, -2731, -55146]\) | \(2251439055699625/25088\) | \(25088\) | \([2]\) | \(6\) | \(0.41310\) | |
14.a4 | 14a6 | \([1, 0, 1, -11, 12]\) | \(128787625/98\) | \(98\) | \([6]\) | \(6\) | \(-0.68551\) |
Rank
sage: E.rank()
The elliptic curves in class 14a have rank \(0\).
Complex multiplication
The elliptic curves in class 14a do not have complex multiplication.Modular form 14.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 3 & 6 & 6 \\ 2 & 1 & 6 & 6 & 3 & 3 \\ 3 & 6 & 1 & 9 & 2 & 18 \\ 3 & 6 & 9 & 1 & 18 & 2 \\ 6 & 3 & 2 & 18 & 1 & 9 \\ 6 & 3 & 18 & 2 & 9 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.