Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+42x+37\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+42xz^2+37z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3375x+37125\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7, 25)$ | $0.82585659471449426692664928051$ | $\infty$ |
Integral points
\((7,\pm 25)\)
Invariants
| Conductor: | $N$ | = | \( 2300 \) | = | $2^{2} \cdot 5^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-5750000$ | = | $-1 \cdot 2^{4} \cdot 5^{6} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{32000}{23} \) | = | $2^{8} \cdot 5^{3} \cdot 23^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.014550364498581519235163768062$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0503183809022801430079541418$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7198241144476085$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.9458333541728505$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.82585659471449426692664928051$ |
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| Real period: | $\Omega$ | ≈ | $1.5251738626275913511807901925$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5191497850743489292476244858 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.519149785 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.525174 \cdot 0.825857 \cdot 2}{1^2} \\ & \approx 2.519149785\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 288 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $23$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 576 & 535 \\ 115 & 231 \end{array}\right),\left(\begin{array}{rr} 511 & 420 \\ 15 & 571 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 685 & 6 \\ 684 & 7 \end{array}\right),\left(\begin{array}{rr} 413 & 0 \\ 0 & 689 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[690])$ is a degree-$2308331520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/690\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 575 = 5^{2} \cdot 23 \) |
| $5$ | additive | $14$ | \( 92 = 2^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 100 = 2^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 2300.c
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 92.b2, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.23.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.12167.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.15111414000.4 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.66125.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.2313060765625.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.242714241693094579272000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.1825454379007169699924376000000000.1 | \(\Z/6\Z\) | 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 | add | ord | add | ord | ss | ord | ord | ord | split | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | - | 1 | 3,1 | 1 | 1 | 1 | 2 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.