Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-16188453x+25008455133\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-16188453xz^2+25008455133z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-20980235763x+1167109186218318\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(2078, 17595\right)\) | \(\left(2753, 34902\right)\) |
$\hat{h}(P)$ | ≈ | $2.0277052527298331320756286092$ | $3.8203491399332156700965439016$ |
Torsion generators
\( \left(\frac{9643}{4}, -\frac{9643}{8}\right) \)
Integral points
\( \left(-1354, 211503\right) \), \( \left(-1354, -210149\right) \), \( \left(406, 135823\right) \), \( \left(406, -136229\right) \), \( \left(2078, 17595\right) \), \( \left(2078, -19673\right) \), \( \left(2753, 34902\right) \), \( \left(2753, -37655\right) \), \( \left(6071, 384951\right) \), \( \left(6071, -391022\right) \)
Invariants
Conductor: | \( 451770 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 37^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1187970282688438790640 $ | = | $2^{4} \cdot 3^{3} \cdot 5 \cdot 11^{8} \cdot 37^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{182864522286982801}{463015182960} \) | = | $2^{-4} \cdot 3^{-3} \cdot 5^{-1} \cdot 11^{-8} \cdot 567601^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.9206449896150698182739882476\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.1151860332929575960899404121\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0750149448569075\dots$ | |||
Szpiro ratio: | $4.7164861960182565\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $7.6601680556093893434175521190\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.15436743496945459873016678136\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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Tamagawa product: | $ 32 $ = $ 2\cdot1\cdot1\cdot2^{3}\cdot2 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 9.4598439534350071024540380479 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 9.459843953 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.154367 \cdot 7.660168 \cdot 32}{2^2} \approx 9.459843953$
Modular invariants
Modular form 451770.2.a.c
For more coefficients, see the Downloads section to the right.
Modular degree: | 38928384 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$11$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$37$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
---|---|---|
$2$ | 2B | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 97680 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 44879 & 0 \\ 0 & 97679 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 97676 & 97677 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 35521 & 34336 \\ 57128 & 79329 \end{array}\right),\left(\begin{array}{rr} 82178 & 94387 \\ 84175 & 14542 \end{array}\right),\left(\begin{array}{rr} 22016 & 29045 \\ 31635 & 10546 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 97582 & 97667 \end{array}\right),\left(\begin{array}{rr} 79736 & 44881 \\ 9583 & 13210 \end{array}\right),\left(\begin{array}{rr} 97665 & 16 \\ 97664 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 34336 \\ 90428 & 6069 \end{array}\right)$.
The torsion field $K:=\Q(E[97680])$ is a degree-$70934367043584000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/97680\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 451770c
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330c4, its twist by $37$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.