Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2+4278x+15614\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z+4278xz^2+15614z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+68445x+1067742\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{25}{9}, \frac{4441}{27}\right)\) |
$\hat{h}(P)$ | ≈ | $3.8591653925831260945758401293$ |
Integral points
None
Invariants
Conductor: | \( 27378 \) | = | $2 \cdot 3^{4} \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-5130328403538 $ | = | $-1 \cdot 2 \cdot 3^{12} \cdot 13^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{3375}{2} \) | = | $2^{-1} \cdot 3^{3} \cdot 5^{3}$ | 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: | $1.1281938274148067430025302182\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.2528931399840713164194587395\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.4265653296335434\dots$ | |||
Szpiro ratio: | $3.5916038159993846\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $3.8591653925831260945758401293\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.46704900549047514533265386624\dots$ | 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: | $ 2 $ = $ 1\cdot1\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: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
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) $ ≈ $ 3.6048387172584162581022780271 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 3.604838717 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.467049 \cdot 3.859165 \cdot 2}{1^2} \approx 3.604838717$
Modular invariants
Modular form 27378.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 41472 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$3$ | $1$ | $II^{*}$ | Additive | 1 | 4 | 12 | 0 |
$13$ | $2$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2G | 8.2.0.1 |
$3$ | 3B | 3.4.0.1 |
$7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 4537 & 2016 \\ 4536 & 4537 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 5460 & 1 \end{array}\right),\left(\begin{array}{rr} 1093 & 546 \\ 5733 & 547 \end{array}\right),\left(\begin{array}{rr} 4369 & 2184 \\ 4368 & 2185 \end{array}\right),\left(\begin{array}{rr} 1 & 546 \\ 0 & 5617 \end{array}\right),\left(\begin{array}{rr} 5511 & 2366 \\ 5096 & 2391 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6006 & 1 \end{array}\right),\left(\begin{array}{rr} 3823 & 546 \\ 6279 & 2731 \end{array}\right),\left(\begin{array}{rr} 1 & 2184 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6474 \\ 546 & 3277 \end{array}\right),\left(\begin{array}{rr} 6007 & 1794 \\ 5460 & 3823 \end{array}\right),\left(\begin{array}{rr} 3550 & 1833 \\ 5733 & 2185 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2016 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 3744 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5825 & 2912 \\ 3640 & 2913 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2184 & 1 \end{array}\right),\left(\begin{array}{rr} 4535 & 0 \\ 0 & 6551 \end{array}\right)$.
The torsion field $K:=\Q(E[6552])$ is a degree-$410847510528$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6552\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 27378.f
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162.b4, its twist by $-39$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-39}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.648.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.25625808.3 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.14414517.1 | \(\Z/7\Z\) | Not in database |
$6$ | 6.0.2767587264.5 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.5910138320875776.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | 12.0.1870004703089601.1 | \(\Z/21\Z\) | Not in database |
$18$ | 18.0.891136206059071970130118099934976.1 | \(\Z/9\Z\) | Not in database |
$18$ | 18.2.146523640191573076809272449302528.1 | \(\Z/6\Z\) | Not in database |
$18$ | 18.6.785127530176038863218409472.1 | \(\Z/14\Z\) | Not in database |
$18$ | 18.6.8943093273411442676347195392.3 | \(\Z/21\Z\) | 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 | add | ss | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 6 | - | 1,1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 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.