Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+2025x+58375\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+2025xz^2+58375z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2623725x+2684184750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-21, 95)$ | $0.88602288337548506424197309138$ | $\infty$ |
Integral points
\( \left(-21, 95\right) \), \( \left(-21, -74\right) \), \( \left(421, 8493\right) \), \( \left(421, -8914\right) \)
Invariants
| Conductor: | $N$ | = | \( 8450 \) | = | $2 \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1960891156250$ | = | $-1 \cdot 2 \cdot 5^{6} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{12167}{26} \) | = | $2^{-1} \cdot 13^{-1} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0429677724130855197050886612$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0442258625347330356220347262$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8441470072585473$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.919887694020949$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.88602288337548506424197309138$ |
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| Real period: | $\Omega$ | ≈ | $0.57554471791414890180911220021$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.0397831619112976074092639553 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.039783162 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.575545 \cdot 0.886023 \cdot 4}{1^2} \\ & \approx 2.039783162\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12096 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $13$ | $4$ | $I_{1}^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 946 & 945 \\ 1395 & 3736 \end{array}\right),\left(\begin{array}{rr} 4663 & 18 \\ 4662 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 3164 & 3735 \\ 1305 & 3764 \end{array}\right),\left(\begin{array}{rr} 3521 & 4230 \\ 4050 & 541 \end{array}\right),\left(\begin{array}{rr} 3743 & 0 \\ 0 & 4679 \end{array}\right),\left(\begin{array}{rr} 3286 & 945 \\ 225 & 3736 \end{array}\right)$.
The torsion field $K:=\Q(E[4680])$ is a degree-$521711124480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4680\Z)$.
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 | $4$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $14$ | \( 338 = 2 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 8450c
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 26a3, its twist by $65$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/3\Z\) | 2.2.65.1-52.1-j3 |
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.20049822000.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.46411625.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.2.17576000.1 | \(\Z/6\Z\) | 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 |
| $12$ | 12.0.2378670782436000000.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.132053982536839773556703232000000000.2 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.26207177223214467584000000000.1 | \(\Z/18\Z\) | 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 | ord | add | ord | ord | add | ord | ord | ss | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | - | 1 | 1 | - | 1 | 1 | 3,1 | 1 | 1 | 1 | 3,1 | 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.