Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4032.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4032.e1 | 4032i3 | \([0, 0, 0, -774156, -262174736]\) | \(268498407453697/252\) | \(48157949952\) | \([2]\) | \(24576\) | \(1.7781\) | |
4032.e2 | 4032i5 | \([0, 0, 0, -526476, 145621744]\) | \(84448510979617/933897762\) | \(178470641597939712\) | \([2]\) | \(49152\) | \(2.1247\) | |
4032.e3 | 4032i4 | \([0, 0, 0, -59916, -1997840]\) | \(124475734657/63011844\) | \(12041750911647744\) | \([2, 2]\) | \(24576\) | \(1.7781\) | |
4032.e4 | 4032i2 | \([0, 0, 0, -48396, -4094480]\) | \(65597103937/63504\) | \(12135803387904\) | \([2, 2]\) | \(12288\) | \(1.4316\) | |
4032.e5 | 4032i1 | \([0, 0, 0, -2316, -94736]\) | \(-7189057/16128\) | \(-3082108796928\) | \([2]\) | \(6144\) | \(1.0850\) | \(\Gamma_0(N)\)-optimal |
4032.e6 | 4032i6 | \([0, 0, 0, 222324, -15432464]\) | \(6359387729183/4218578658\) | \(-806182936033886208\) | \([2]\) | \(49152\) | \(2.1247\) |
Rank
sage: E.rank()
The elliptic curves in class 4032.e have rank \(0\).
Complex multiplication
The elliptic curves in class 4032.e do not have complex multiplication.Modular form 4032.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.