Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2+7870x-667128\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z+7870xz^2-667128z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+125925x-42570250\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(74, 525\right)\) | \(\left(84, 720\right)\) |
$\hat{h}(P)$ | ≈ | $1.4814771797731428738016419108$ | $3.4414473320875884629325533109$ |
Torsion generators
\( \left(59, -30\right) \)
Integral points
\( \left(59, -30\right) \), \( \left(74, 525\right) \), \( \left(74, -600\right) \), \( \left(84, 720\right) \), \( \left(84, -805\right) \), \( \left(315, 5586\right) \), \( \left(315, -5902\right) \), \( \left(788, 21840\right) \), \( \left(788, -22629\right) \), \( \left(1274, 44925\right) \), \( \left(1274, -46200\right) \), \( \left(17399, 2286300\right) \), \( \left(17399, -2303700\right) \)
Invariants
Conductor: | \( 26775 \) | = | $3^{2} \cdot 5^{2} \cdot 7 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-222333324609375 $ | = | $-1 \cdot 3^{14} \cdot 5^{8} \cdot 7 \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{4733169839}{19518975} \) | = | $3^{-8} \cdot 5^{-2} \cdot 7^{-1} \cdot 17^{-1} \cdot 23^{3} \cdot 73^{3}$ | 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: | $1.4355547739356544224373376233\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.081529673384549389439335338226\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8766842762334948\dots$ | |||
Szpiro ratio: | $3.9566283395067545\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $4.6552241952899581019758333377\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.28342632022502529397717116410\dots$ | 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: | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot1 $ | 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])
|
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! $ ≈ $ 5.2776522539741494034878322390 $ | 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
$\displaystyle 5.277652254 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.283426 \cdot 4.655224 \cdot 16}{2^2} \approx 5.277652254$
Modular invariants
Modular form 26775.2.a.p
For more coefficients, see the Downloads section to the right.
Modular degree: | 98304 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{8}^{*}$ | Additive | -1 | 2 | 14 | 8 |
$5$ | $4$ | $I_{2}^{*}$ | Additive | 1 | 2 | 8 | 2 |
$7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$17$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 | 16.24.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 28560 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 10088 & 1 \\ 6799 & 10 \end{array}\right),\left(\begin{array}{rr} 5699 & 28544 \\ 17392 & 315 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10721 & 16 \\ 3506 & 21327 \end{array}\right),\left(\begin{array}{rr} 12256 & 5 \\ 24435 & 28546 \end{array}\right),\left(\begin{array}{rr} 28545 & 16 \\ 28544 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 7148 & 7269 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 28462 & 28547 \end{array}\right),\left(\begin{array}{rr} 19039 & 28544 \\ 9512 & 28431 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 28556 & 28557 \end{array}\right)$.
The torsion field $K:=\Q(E[28560])$ is a degree-$465740884869120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/28560\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 26775bq
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1785m1, its twist by $-15$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-119}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{1785}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{-119})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{21})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{-51})\) | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.10152029750625.7 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/16\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | ord | add | add | split | ord | ord | split | ord | ss | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 8 | - | - | 3 | 2 | 2 | 3 | 2 | 4,4 | 2 | 2 | 2 | 2 | 2 | 2,2 |
$\mu$-invariant(s) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.