Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-19999x+70995122\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-19999xz^2+70995122z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-25918083x+3312426177918\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-329, 6644\right)\) |
$\hat{h}(P)$ | ≈ | $0.50883903100376855956652449301$ |
Torsion generators
\( \left(-\frac{1721}{4}, \frac{1717}{8}\right) \)
Integral points
\( \left(-329, 6644\right) \), \( \left(-329, -6316\right) \), \( \left(-104, 8534\right) \), \( \left(-104, -8431\right) \), \( \left(1210, 42041\right) \), \( \left(1210, -43252\right) \), \( \left(4855, 335828\right) \), \( \left(4855, -340684\right) \)
Invariants
Conductor: | \( 25230 \) | = | $2 \cdot 3 \cdot 5 \cdot 29^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-2177003129753318400 $ | = | $-1 \cdot 2^{10} \cdot 3^{20} \cdot 5^{2} \cdot 29^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{36267977929301}{89261680665600} \) | = | $-1 \cdot 2^{-10} \cdot 3^{-20} \cdot 5^{-2} \cdot 79^{3} \cdot 419^{3}$ | 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: | $2.1976632440304927779511011976\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.3558392865338742711552831895\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.2035853381343617\dots$ | |||
Szpiro ratio: | $4.901388115415945\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.50883903100376855956652449301\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.20919683690680219451878769330\dots$ | 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: | $ 160 $ = $ 2\cdot( 2^{2} \cdot 5 )\cdot2\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'(E,1) $ ≈ $ 4.2579006312284254683450093876 $ | 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
$\displaystyle 4.257900631 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.209197 \cdot 0.508839 \cdot 160}{2^2} \approx 4.257900631$
Modular invariants
Modular form 25230.2.a.h
For more coefficients, see the Downloads section to the right.
Modular degree: | 448000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{10}$ | Non-split multiplicative | 1 | 1 | 10 | 10 |
$3$ | $20$ | $I_{20}$ | Split multiplicative | -1 | 1 | 20 | 20 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$29$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 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 | 2.3.0.1 |
$5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 580 = 2^{2} \cdot 5 \cdot 29 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 62 & 15 \\ 365 & 318 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 345 & 46 \end{array}\right),\left(\begin{array}{rr} 561 & 20 \\ 560 & 21 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 340 & 231 \end{array}\right),\left(\begin{array}{rr} 117 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[580])$ is a degree-$109132800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/580\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 25230i
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-29}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.2.2438900.2 | \(\Z/4\Z\) | Not in database |
$4$ | 4.4.3048625.2 | \(\Z/10\Z\) | Not in database |
$8$ | 8.0.60909908070400.23 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.95171731360000.28 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$8$ | 8.0.2379293284000000.12 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$8$ | 8.4.148705830250000.8 | \(\Z/20\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | Not in database |
$20$ | 20.0.125571223144625902073297766037285327911376953125.1 | \(\Z/10\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | split | nonsplit | ord | ss | ord | ord | ord | ord | add | ss | ord | ss | ord | ord |
$\lambda$-invariant(s) | 2 | 2 | 1 | 3 | 1,1 | 1 | 1 | 3 | 1 | - | 1,1 | 1 | 1,1 | 1 | 1 |
$\mu$-invariant(s) | 1 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | - | 0,0 | 0 | 0,0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.