Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1516307x-709369549\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1516307xz^2-709369549z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-24260907x-45423912026\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(4059, 242914\right) \) | $2.0252365462190857904935600967$ | $\infty$ |
| \( \left(-645, 322\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([4059:242914:1]\) | $2.0252365462190857904935600967$ | $\infty$ |
| \([-645:322:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(16235, 1959552\right) \) | $2.0252365462190857904935600967$ | $\infty$ |
| \( \left(-2581, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-645, 322\right) \), \( \left(4059, 242914\right) \), \( \left(4059, -246974\right) \)
\([-645:322:1]\), \([4059:242914:1]\), \([4059:-246974:1]\)
\( \left(-2581, 0\right) \), \((16235,\pm 1959552)\)
Invariants
| Conductor: | $N$ | = | \( 75690 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 29^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $5504323962635550720$ | = | $2^{20} \cdot 3^{16} \cdot 5 \cdot 29^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{21685195471991381}{309586821120} \) | = | $2^{-20} \cdot 3^{-10} \cdot 5^{-1} \cdot 11^{3} \cdot 101^{3} \cdot 251^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4003957980845749689401077553$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0092656962539016164466671287$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0328645517916797$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.8341665756404435$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0252365462190857904935600967$ |
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| Real period: | $\Omega$ | ≈ | $0.13616898818229940878442687412$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ ( 2^{2} \cdot 5 )\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.030976453138702536041682004 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.030976453 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.136169 \cdot 2.025237 \cdot 160}{2^2} \\ & \approx 11.030976453\end{aligned}$$
Modular invariants
Modular form 75690.2.a.bo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1792000 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $3$ | $4$ | $I_{10}^{*}$ | additive | -1 | 2 | 16 | 10 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $2$ | $III$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $5$ | 5B | 5.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 1222 & 15 \\ 945 & 898 \end{array}\right),\left(\begin{array}{rr} 1159 & 0 \\ 0 & 1739 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1282 & 1173 \\ 495 & 976 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1500 & 1391 \end{array}\right),\left(\begin{array}{rr} 1721 & 20 \\ 1720 & 21 \end{array}\right),\left(\begin{array}{rr} 871 & 600 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1740])$ is a degree-$5238374400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1740\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1305 = 3^{2} \cdot 5 \cdot 29 \) |
| $3$ | additive | $8$ | \( 8410 = 2 \cdot 5 \cdot 29^{2} \) |
| $5$ | split multiplicative | $6$ | \( 7569 = 3^{2} \cdot 29^{2} \) |
| $29$ | additive | $226$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 75690.bo
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 25230.h1, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{145}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{87 -12 \sqrt{-29}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-870 -6 \sqrt{145}})\) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7708910240160000.57 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $20$ | 20.4.7414855155467014891526159786735661327838897705078125.1 | \(\Z/10\Z\) | 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 | split | add | split | ord | ss | ord | ord | ord | ord | add | ss | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 7 | - | 2 | 1 | 1,1 | 1 | 1 | 3 | 1 | - | 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 |
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.