Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-50251x+1521398\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-50251xz^2+1521398z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-65124675x+71177730750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-38, 1856)$ | $0.84563212296501937979072103824$ | $\infty$ |
| $(207, -104)$ | $0$ | $2$ |
Integral points
\( \left(-113, 2456\right) \), \( \left(-113, -2344\right) \), \( \left(-38, 1856\right) \), \( \left(-38, -1819\right) \), \( \left(207, -104\right) \), \( \left(232, 1421\right) \), \( \left(232, -1654\right) \), \( \left(3343, 191192\right) \), \( \left(3343, -194536\right) \)
Invariants
| Conductor: | $N$ | = | \( 7350 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $7115411520000000$ | = | $2^{12} \cdot 3^{3} \cdot 5^{7} \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{7633736209}{3870720} \) | = | $2^{-12} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{3} \cdot 179^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7342467188866314435641438991$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.043427311858075396288912139236$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9780810097967987$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.952333785685826$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.84563212296501937979072103824$ |
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| Real period: | $\Omega$ | ≈ | $0.37051062413689921157474990892$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot3\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.7597882280397651779345539276 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.759788228 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.370511 \cdot 0.845632 \cdot 48}{2^2} \\ & \approx 3.759788228\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 55296 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | 4.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 16 & 423 \\ 521 & 194 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 374 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 344 & 819 \\ 645 & 374 \end{array}\right),\left(\begin{array}{rr} 217 & 24 \\ 702 & 7 \end{array}\right),\left(\begin{array}{rr} 817 & 24 \\ 816 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 488 & 837 \\ 123 & 86 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 249 & 388 \\ 500 & 629 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | split multiplicative | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 7350w
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210a1, its twist by $-35$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-35}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-35})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{21})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-7})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.1418090625.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.343064484000000.20 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.112021056000000.6 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.2710632960000.28 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.121550625.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 |
| $18$ | 18.0.99742940201137293350060715000000000000.1 | \(\Z/18\Z\) | 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 | split | add | add | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 4 | - | - | 1,1 | 1 | 1 | 1 | 3,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-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.