Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+239024x-10823394\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+239024xz^2-10823394z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+309775725x-505905586002\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{19825}{441}, \frac{852167}{9261}\right)\) |
$\hat{h}(P)$ | ≈ | $7.4646758567441905128452464826$ |
Torsion generators
\( \left(234, 7494\right) \)
Integral points
\( \left(234, 7494\right) \), \( \left(234, -7729\right) \)
Invariants
Conductor: | \( 30446 \) | = | $2 \cdot 13 \cdot 1171$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-924784759817371648 $ | = | $-1 \cdot 2^{18} \cdot 13^{3} \cdot 1171^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1510256987478812234375}{924784759817371648} \) | = | $2^{-18} \cdot 5^{6} \cdot 7^{3} \cdot 13^{-3} \cdot 53^{3} \cdot 1171^{-3} \cdot 1237^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.1365387016127155241593956750\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.1365387016127155241593956750\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9957987952611158\dots$ | |||
Szpiro ratio: | $4.723744394634306\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $7.4646758567441905128452464826\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.16182151222912307147803757406\dots$ | 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);
|
Tamagawa product: | $ 18 $ = $ 2\cdot3\cdot3 $ | 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: | $3$ | 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) $ ≈ $ 2.4158902708771395317797074744 $ | 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
$\displaystyle 2.415890271 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.161822 \cdot 7.464676 \cdot 18}{3^2} \approx 2.415890271$
Modular invariants
Modular form 30446.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 334368 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{18}$ | Non-split multiplicative | 1 | 1 | 18 | 18 |
$13$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$1171$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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 |
---|---|---|
$3$ | 3Cs.1.1 | 3.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 274014 = 2 \cdot 3^{2} \cdot 13 \cdot 1171 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 265831 & 18 \\ 157815 & 273667 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 168625 & 18 \\ 147555 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 243569 \end{array}\right),\left(\begin{array}{rr} 273997 & 18 \\ 273996 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[274014])$ is a degree-$7976366941537612800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/274014\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 30446.a
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$3$ | 3.1.15223.1 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.3527773894567.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.6256972683.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$9$ | 9.3.244958648221341647872568535987.1 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.1620127962137529153006436479826094075951180881657405879732563.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.50694006747301596338301854967121679019558066456267.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.458073824236380006330753539976544760253971187.1 | \(\Z/3\Z \oplus \Z/9\Z\) | 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 | 1171 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | ord | ss | ord | ord | split | ord | ord | ord | ss | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | 4 | 1 | 1,1 | 1 | 1 | 4 | 1 | 1 | 3 | 1,1 | 1 | 1 | 1 | 1 | 1 | ? |
$\mu$-invariant(s) | 0 | 1 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | ? |
An entry ? indicates that the invariants have not yet been computed.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.