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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 181202.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
181202.d1 | 181202a6 | \([1, 0, 0, -247387918, -1497689606652]\) | \(2251439055699625/25088\) | \(18657996813433983488\) | \([2]\) | \(22643712\) | \(3.2667\) | |
181202.d2 | 181202a5 | \([1, 0, 0, -15449358, -23441731580]\) | \(-548347731625/1835008\) | \(-1364699195496885649408\) | \([2]\) | \(11321856\) | \(2.9201\) | |
181202.d3 | 181202a4 | \([1, 0, 0, -3218223, -1821705551]\) | \(4956477625/941192\) | \(699966411703984286792\) | \([2]\) | \(7547904\) | \(2.7174\) | |
181202.d4 | 181202a2 | \([1, 0, 0, -953198, 357882706]\) | \(128787625/98\) | \(72882800052476498\) | \([2]\) | \(2515968\) | \(2.1680\) | |
181202.d5 | 181202a1 | \([1, 0, 0, -47188, 7981644]\) | \(-15625/28\) | \(-20823657157850428\) | \([2]\) | \(1257984\) | \(1.8215\) | \(\Gamma_0(N)\)-optimal |
181202.d6 | 181202a3 | \([1, 0, 0, 405817, -166968887]\) | \(9938375/21952\) | \(-16325747211754735552\) | \([2]\) | \(3773952\) | \(2.3708\) |
Rank
sage: E.rank()
The elliptic curves in class 181202.d have rank \(0\).
Complex multiplication
The elliptic curves in class 181202.d do not have complex multiplication.Modular form 181202.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.