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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-231173x-28557902347\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-231173xz^2-28557902347z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-299599587x-1332396593091234\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 0, 1, -231173, -28557902347])
 
Copy content gp:E = ellinit([1, 0, 1, -231173, -28557902347])
 
Copy content magma:E := EllipticCurve([1, 0, 1, -231173, -28557902347]);
 
Copy content oscar:E = elliptic_curve([1, 0, 1, -231173, -28557902347])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

$P$$\hat{h}(P)$Order
\( \left(\frac{318669}{16}, \frac{178876945}{64}\right) \)$10.038730987442165340865073275$$\infty$
\( \left(\frac{12327}{4}, -\frac{12331}{8}\right) \)$0$$2$

$P$$\hat{h}(P)$Order
\([1274676:178876945:64]\)$10.038730987442165340865073275$$\infty$
\([24654:-12331:8]\)$0$$2$

$P$$\hat{h}(P)$Order
\( \left(\frac{2868033}{4}, \frac{4846886505}{8}\right) \)$10.038730987442165340865073275$$\infty$
\( \left(110946, 0\right) \)$0$$2$

Integral points

None

Copy content comment:Integral points
 
Copy content sage:E.integral_points()
 
Copy content magma:IntegralPoints(E);
 

Invariants

Conductor: $N$  =  \( 25215 \) = $3 \cdot 5 \cdot 41^{2}$
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Minimal Discriminant: $\Delta$  =  $-352317969757428581297025$ = $-1 \cdot 3^{16} \cdot 5^{2} \cdot 41^{9} $
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: $j$  =  \( -\frac{4173281}{1076168025} \) = $-1 \cdot 3^{-16} \cdot 5^{-2} \cdot 7^{3} \cdot 23^{3}$
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $3.1971208982063797227458489136$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $0.41194184817814886984577638382$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.1547638000559055$
Szpiro ratio: $\sigma_{m}$ ≈ $6.0850873897323075$
Intrinsic torsion order: $\#E(\mathbb Q)_\text{tors}^\text{is}$ = $1$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 1$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 1$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $10.038730987442165340865073275$
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: $\Omega$ ≈ $0.043816859074573233399394398271$
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: $\prod_{p}c_p$ = $ 64 $  = $ 2^{4}\cdot2\cdot2 $
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: $\#E(\Q)_{\mathrm{tor}}$ = $2$
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: $ L'(E,1)$ ≈ $7.0378505754288761336900711336 $
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 7.037850575 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043817 \cdot 10.038731 \cdot 64}{2^2} \\ & \approx 7.037850575\end{aligned}$$

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, 0, 1, -231173, -28557902347]); r = E.rank(); ar = E.analytic_rank(); assert r == ar; Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical(); omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order(); assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, 0, 1, -231173, -28557902347]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar; sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1); reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E); assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form   25215.2.a.i

\( q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{10} - q^{12} + 4 q^{13} - 4 q^{14} + q^{15} - q^{16} - 4 q^{17} + q^{18} + O(q^{20}) \) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 3358720
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

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 $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$3$ $16$ $I_{16}$ split multiplicative -1 1 16 16
$5$ $2$ $I_{2}$ split multiplicative -1 1 2 2
$41$ $2$ $III^{*}$ additive 1 2 9 0

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

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 8.12.0.27 $12$

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[5, 2, 1334, 9], [3, 8, 1632, 1619], [48, 3, 357, 1624], [1, 0, 8, 1], [5, 8, 48, 77], [1, 418, 410, 1], [1633, 8, 1632, 9], [1236, 429, 813, 388], [1, 8, 0, 1], [1, 4, 4, 17]] GL(2,Integers(1640)).subgroup(gens)
 
Copy content magma:Gens := [[5, 2, 1334, 9], [3, 8, 1632, 1619], [48, 3, 357, 1624], [1, 0, 8, 1], [5, 8, 48, 77], [1, 418, 410, 1], [1633, 8, 1632, 9], [1236, 429, 813, 388], [1, 8, 0, 1], [1, 4, 4, 17]]; sub<GL(2,Integers(1640))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1640 = 2^{3} \cdot 5 \cdot 41 \), index $48$, genus $1$, and generators

$\left(\begin{array}{rr} 5 & 2 \\ 1334 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 1632 & 1619 \end{array}\right),\left(\begin{array}{rr} 48 & 3 \\ 357 & 1624 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 1 & 418 \\ 410 & 1 \end{array}\right),\left(\begin{array}{rr} 1633 & 8 \\ 1632 & 9 \end{array}\right),\left(\begin{array}{rr} 1236 & 429 \\ 813 & 388 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[1640])$ is a degree-$42319872000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1640\Z)$.

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
$2$ good $2$ \( 41 \)
$3$ split multiplicative $4$ \( 8405 = 5 \cdot 41^{2} \)
$5$ split multiplicative $6$ \( 5043 = 3 \cdot 41^{2} \)
$41$ additive $482$ \( 15 = 3 \cdot 5 \)

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 25215.i consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 25215.h2, its twist by $41$.

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
$2$ \(\Q(\sqrt{-41}) \) \(\Z/2\Z \oplus \Z/2\Z\) not in database
$4$ \(\Q(\sqrt[4]{41})\) \(\Z/4\Z\) not in database
$8$ 8.0.1216026685696.2 \(\Z/2\Z \oplus \Z/4\Z\) 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
$8$ 8.0.30400667142400.9 \(\Z/8\Z\) 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 \(\Z/2\Z \oplus \Z/6\Z\) 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 ord split split ord ss ord ord ss ss ord ord ord add ord ord
$\lambda$-invariant(s) 6 4 2 1 1,1 1 1 1,3 1,1 1 1 1 - 1 1
$\mu$-invariant(s) 1 0 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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.