Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-584180454x+834238847475\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-584180454xz^2+834238847475z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-757097868411x+38933604135828054\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Torsion generators
\( \left(56001, 11961531\right) \)
Integral points
\( \left(56001, 11961531\right) \), \( \left(56001, -12017533\right) \)
Invariants
Conductor: | \( 18354 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $12458301538998671409274874352 $ | = | $2^{4} \cdot 3^{6} \cdot 7 \cdot 19^{16} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{22047775488403890529761445244257}{12458301538998671409274874352} \) | = | $2^{-4} \cdot 3^{-6} \cdot 7^{-1} \cdot 19^{-16} \cdot 23^{-2} \cdot 28040661793^{3}$ | 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: | $4.0810227591363038739952877516\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $4.0810227591363038739952877516\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0599914760700624\dots$ | |||
Szpiro ratio: | $7.3511595318762035\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.034476434268761597067528001663\dots$ | 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: | $ 256 $ = $ 2^{2}\cdot2\cdot1\cdot2^{4}\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: | $4$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 0.55162294830018555308044802660 $ | 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
$\displaystyle 0.551622948 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.034476 \cdot 1.000000 \cdot 256}{4^2} \approx 0.551622948$
Modular invariants
Modular form 18354.2.a.p
For more coefficients, see the Downloads section to the right.
Modular degree: | 16171008 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$3$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$19$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
$23$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 | 16.48.0.27 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2576 = 2^{4} \cdot 7 \cdot 23 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 2478 & 2563 \end{array}\right),\left(\begin{array}{rr} 2365 & 16 \\ 192 & 2261 \end{array}\right),\left(\begin{array}{rr} 2561 & 16 \\ 2560 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 2572 & 2573 \end{array}\right),\left(\begin{array}{rr} 1480 & 1 \\ 447 & 10 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1618 & 1939 \\ 991 & 1310 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 652 & 773 \end{array}\right)$.
The torsion field $K:=\Q(E[2576])$ is a degree-$68942168064$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2576\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 18354.p
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{23}) \) | \(\Z/8\Z\) | Not in database |
$2$ | \(\Q(\sqrt{161}) \) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{7}, \sqrt{23})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$4$ | 4.4.12264336.1 | \(\Z/16\Z\) | Not in database |
$8$ | 8.0.624529833984.16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/16\Z\) | Not in database |
$8$ | 8.8.7370282938523904.3 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$8$ | deg 8 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/32\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | 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 | 7 | 19 | 23 |
---|---|---|---|---|---|
Reduction type | split | nonsplit | nonsplit | split | split |
$\lambda$-invariant(s) | 4 | 0 | 0 | 3 | 3 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.