Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 418950b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
418950.b6 | 418950b1 | \([1, -1, 0, -86000742, 306993890916]\) | \(52492168638015625/293197968\) | \(392913318756908250000\) | \([2]\) | \(63700992\) | \(3.1440\) | \(\Gamma_0(N)\)-optimal |
418950.b5 | 418950b2 | \([1, -1, 0, -87544242, 295403749416]\) | \(55369510069623625/3916046302812\) | \(5247876578884009878937500\) | \([2]\) | \(127401984\) | \(3.4906\) | |
418950.b4 | 418950b3 | \([1, -1, 0, -123210117, 16211555541]\) | \(154357248921765625/89242711068672\) | \(119593768060683782208000000\) | \([2]\) | \(191102976\) | \(3.6933\) | |
418950.b3 | 418950b4 | \([1, -1, 0, -1333314117, -18676264932459]\) | \(195607431345044517625/752875610010048\) | \(1008925322907352937247000000\) | \([2]\) | \(382205952\) | \(4.0399\) | |
418950.b2 | 418950b5 | \([1, -1, 0, -6743667492, -213150352753584]\) | \(25309080274342544331625/191933498523648\) | \(257209244661445558272000000\) | \([2]\) | \(573308928\) | \(4.2426\) | |
418950.b1 | 418950b6 | \([1, -1, 0, -107898483492, -13641755641201584]\) | \(103665426767620308239307625/5961940992\) | \(7989571133042688000000\) | \([2]\) | \(1146617856\) | \(4.5892\) |
Rank
sage: E.rank()
The elliptic curves in class 418950b have rank \(1\).
Complex multiplication
The elliptic curves in class 418950b do not have complex multiplication.Modular form 418950.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.